math.FA — Pith

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math.FA 2026-05-18 2 theorems

Residual collapse equates ordered POVM realizations by surviving effects

by James Tian

Different orderings and couplings reduce to the same canonical form whose non-escape coordinates are orthogonal and sum to the identity.

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Ordered realizations of discrete POVMs are studied through a residual transform generated by sequential tests. One application of the transform replaces each coordinate by the effect obtained after all earlier tests have failed, and appends the remaining mass as a terminal outcome. Under natural hypotheses, iterating the transform produces a collapsed POVM whose non-escape coordinates are the parts of the original effects that survive all earlier tests. The resulting collapse map gives an equivalence relation on ordered POVM realizations. Its range and fibers are characterized. The range consists of collapsed POVMs, whose non-escape coordinates are mutually orthogonal and whose support projections strongly sum to the identity. The fiber over a collapsed POVM consists of all ordered realizations with the same residually visible compressions. In particular, different ordered realizations, including ones with different off-diagonal coupling data, can have the same collapsed image. After collapse, the non-escape coordinates are fixed under further residual iteration. The remaining dynamics takes place in the escape effect, which is fragmented by a universal scalar functional calculus.

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math.FA 2026-05-18 2 theorems

Zero-measure spectra force unitarity under controlled inverse growth

by Thomas Ransford

A proof of Esterle's conjecture on negative powers of Hilbert-space contractions

For any such thin set E there is a sequence u_n so that slow growth of negative powers implies the contraction is unitary.

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We establish the following result, confirming a conjecture of Jean Esterle. For each closed subset $E$ of the unit circle of Lebesgue measure zero, there exists a positive sequence $u_n\to\infty$ with the following property: if $T$ is a contraction on a Hilbert space such that $\sigma(T)\subset E$ and $\|T^{-n}\|=O(u_n)$ as $n\to\infty$, then $T$ is a unitary operator. A key tool used in the proof is a result generalizing the well-known fact that closed subsets $E$ of the real axis of Lebesgue measure zero are removable for bounded holomorphic functions. We show that such sets remain removable even for certain unbounded holomorphic functions of moderate growth near $E$, where the notion of `moderate' depends on $E$.

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cs.LG 2026-05-15

Operator networks approximate nonlinear maps and their derivatives

by Filippo de Feo

Universal Approximation of Nonlinear Operators and Their Derivatives

Universal approximation theorems cover k-times differentiable operators in weighted Bastiani-Sobolev spaces on general Banach spaces.

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Establishing Universal Approximation Theorems (UATs) for nonlinear operators and their derivatives is a foundational open problem in Operator Learning (OL) and raises delicate questions in Nonlinear Functional Analysis. We prove the first UATs for $k$-times differentiable nonlinear operators and their derivatives via OL architectures, uniformly on compact sets and in weighted Bastiani--Sobolev spaces for general finite input measures. In full Banach-space generality, these are the first complete generalizations of the corresponding influential classical UATs in [Hornik, 1991] to infinite-dimensional spaces and OL and they launch Derivative-Informed Operator Learning (DIOL)-learning nonlinear operators and their derivatives-on general Banach spaces. Based on our UATs, we formulate Bastiani--Sobolev training in DIOL. We present open frontiers where DIOL and our UATs find applications: high-order accuracy in OL; fast constrained optimization in Banach spaces (e.g. optimal control of PDEs, inverse problems) via Learn-Then-Optimize; numerical methods for infinite-dimensional PDEs (e.g. HJB PDEs on Banach spaces from infinite-dimensional optimal control via Optimize-Then-Learn, such as optimal control of PDEs, SPDEs, path-dependent systems, partially observed systems, mean-field control). We parameterize nonlinear operators via Encoder-Decoder Architectures, classical OL architectures. These include DeepONets, Deep-H-ONets, and PCA-Nets, which our UATs cover. Our UATs are based on (i) Approximation Properties of Banach spaces; (ii) continuous Bastiani differentiability (weaker than continuous Fr\'echet differentiability); (iii) $C^k_B$ (Bastiani) compact-open topologies; indeed, UA in $C^k$ (Fr\'echet) compact-open topologies (induced by operator norms) fails; (iv) construction of weighted Bastiani--Sobolev spaces, generalizing classical Gaussian Sobolev spaces on Banach spaces.

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math.AP 2026-07-03

Fractional Dirichlet solutions split into regular and d^a-singular parts

by Gerd Grubb

The structure of solution spaces for fractional-order operators, with gradient estimates

Direct-sum decomposition on C^{1+τ} domains yields gradient estimates in Sobolev and Hölder spaces when a exceeds 1/2

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The solution space of the homogeneous Dirichlet problem for the fractional Laplacian $(-\Delta )^{a}$ ($0<a<1$) or a pseudodifferential generalization $P$, on a bounded open set $\Omega \subset R^n$ with $C^{1+\tau }$-boundary, $$ Pu=f \text{ on }\Omega ,\quad u=0 \text{ on }R^n\setminus \Omega , $$ is analysed in detail. It is shown, both for solutions in Sobolev spaces of Bessel-potential type $H_q^t$ and in H\"older-Zygmund spaces $C_*^t$, that the solution space for $f$ of regularity $s\in [0,\tau -2a)$ is the direct sum of a component $\dot H_q^{2a+s}(\bar\Omega)$ resp. $\dot C_*^{2a+s}(\bar\Omega)$ with full regularity $2a+s$ and a component of the form $d^a$ times a lifting of boundary values by Poisson operators. Here $d(x)=dist(x,\partial\Omega )$. This extends to non-smooth problems results known in the $C^\infty $ setting. The knowledge is used to establish gradient estimates for $a>1/2$, e.g. estimating $d^{1-a+s}\nabla (u/d^a)$ in terms of norms of $f$ and $u$, both in $H_q^t$-spaces and $C_*^t$-spaces. This is entirely new in the case of Bessel-potential spaces; it extends previous results by Fall and Jarohs in H\"older spaces. A new tool is introduced: $\dot H^{s+t}_q(\bar\Omega)\subset d^s\dot H^{t}_q(\bar\Omega)$ holds for $s,t\ge 0$ with $s+t<1+\tau $.

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math.MG 2026-07-03

Minkowski theory resolves Pólya-Szegő capacity conjecture

by Qiushuang Liu, Jie Xiao +3 more

A Minkowski Theory for the Exterior Capacitary Volumes and A Resolution of the P\'olya-Szeg\"o Conjecture

Exterior capacitary volumes satisfy volume-like inequalities that confirm the ball extremizes electrostatic capacity among convex bodies.

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This paper establishes a unified Minkowski theory for exterior p-capacitary volumes and resolves the classical P\'olya-Szeg\"o conjecture on the electrostatic capacity of convex bodies.

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math.FA 2026-07-03

Geometric property on Y ensures all L1(μ,Y) elements have optimal tensor reps

by Luis C. García-Lirola, Juan Guerrero-Viu

Functions in L₁(μ,Y) with optimal tensor representations

Holds for Lipschitz-free spaces over scattered metrics, totally disconnected C(K), and c0(Γ); resolves two projective attainment questions.

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We study the existence and characterization of optimal tensor representations of elements in the space $L_1(\mu,Y)$ of Bochner integrable functions. We completely describe the set of norm-attaining elements in two settings. First, when the Banach space $Y$ is strictly convex, and second, when $Y=L_1(\nu)$ and $\mathbb K=\mathbb R$. In both situations, our analysis yields the existence of non-norm-attaining tensors whenever the underlying measures are not purely atomic. Finally, we introduce a geometric property over $Y$ ensuring that every element in $L_1(\mu, Y)$ admits an optimal representation. In particular, this holds for Lipschitz-free spaces over complete scattered metric spaces, for $C(K)$ spaces when $K$ is a compact Hausdorff totally disconnected space, and for $c_0(\Gamma)$ where $\Gamma$ is any index set. As a byproduct, we settle two open questions regarding projective norm-attainment.

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math.FA 2026-07-03

Counterexample disproves min equality for Daugavet thickness in l1-sums

by Rainis Haller, Andre Ostrak

A counterexample for the Daugavet index of thickness in ell₁-sums

T(X ⊕₁ X) can be strictly smaller than T(X) when X is built from a Daugavet space using a suitable norm.

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We give a negative answer to a question of Haller-Langemets-Lima-Nadel-Rueda Zoca asking whether, for all Banach spaces $X$ and $Y$, the Daugavet index of thickness satisfies \[ T(X\oplus_1 Y)=\min\{T(X),T(Y)\}. \] We show that this equality does hold whenever one of the two summands has the Daugavet property. On the other hand, if $D$ is a Banach space with the Daugavet property and $N$ is a suitable absolute norm, then for $X=D\oplus_N D$, one has $T(X\oplus_1 X)<T(X)$.

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math.GN 2026-07-03

Weak ball of nonseparable Hilbert space homeomorphic to its positive part

by Antonio Avilés

Homeomorphism between close relatives of Hilbertian balls

The same ball is also homeomorphic to its product with the Hilbert cube, and all related B(κ,a,b) spaces coincide up to homeomorphism

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We present a solution to some problems posed by the author and Kalenda. We show that the closed ball of nonseparable Hilbert space in its weak topology is homeomorphic to its positive part, as well as to its product with the Hilbert cube. In the separable setting we obtain that there is a weak homeomorphism of the closed unit ball of $\ell_2$ onto its positive part that preserves the norm, and via a result of Dijkstra and van Mill, the same is true for the ball of $\ell_\infty=\ell_1^*$ in the weak$^*$ topology. All spaces $B(\kappa,a,b)$ considered by the author and Kalenda are shown to be homeomorphic. The solution has been found by AI (Chatgpt 5.5), the role of the author has been to ask the right questions, check and understand the answers, and adapt the writing to his personal human taste.

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math.FA 2026-07-03

Super Delta-points force coordinate rules in Banach sums

by Juan Guerrero-Viu, Joanna Markowicz

On super Delta-points and the convex-DLD2P in absolute sums

Convex-DLD2P passes from absolute sum to factors unless the norm is ell-infinity.

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We partially answer two open questions concerning diameter two properties in absolute sums. First, we identify the conditions that a super $\Delta$-point in an absolute sum of Banach spaces imposes on the coordinates. Secondly, we show that the convex diametral local diameter two property (convex-DLD2P) passes from an absolute sum $X\oplus_N Y$ to its factors whenever $N$ is not the $\ell_\infty$-norm.

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math-ph 2026-07-03

r-deformed Rényi entropy gives tighter Tsallis bound on density operators

by Srikrishna Maity, Shigeru Furuichi +1 more

r-deformed α-z-R\'enyi relative entropy

The three-parameter family lies below an existing upper bound whenever both are applied to quantum states.

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In this article, we consider the $r$-logarithm for defining three-parameter family of R\'{e}nyi relative entropies that are generalization of the $\alpha$-$z$-R\'{e}nyi relative entropies. All the members of $r$-deformed $\alpha$-$z$-R\'{e}nyi relative entropies satisfy the necessary axioms to be a divergence. We expose the range of parameters $\alpha$, $z$ and $r$ for which the data processing inequality holds. We also establish that $r$-deformed $\alpha$-$z$-R\'{e}nyi relative entropy is an upper bound of the Tsallis relative entropy. Now, we have two upper bounds of the Tsallis relative entropy, which are $r$-deformed $\alpha$-$z$-R\'{e}nyi relative entropy and the other one, which is discussed in literature. We investigate the order relationship between these two upper bounds of the Tsallis relative entropy. We observe that our new upper bound is more tighter when applicable to the density operators.

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math.FA 2026-07-03

Every quasinilpotent operator has 1 in its power set

by C. L. Hu, Y. Q. Ji +1 more

On the Power Set of Quasinilpotent Operators in Banach Spaces

Any right-closed subset of [0,1] containing 1 can be realized exactly as Λ(T) on spaces with uniform multiplicity infinity

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For a quasinilpotent operator $T$ on a Banach space $X$, Douglas and Yang defined $k_{x}=\limsup\limits_{\lambda\rightarrow 0}\frac{\ln\|(\lambda-T)^{-1}x\|}{\ln\|(\lambda-T)^{-1}\|}$ for each non-zero vector $x$, and called $\Lambda(T)=\{k_x:x\neq 0\}$ the power set of $T$. In this paper, we prove that $\Lambda(T)$ always contains $1$ for every quasinilpotent operator $T$ on $X$. Moreover, we introduce the concept of a Banach space $X$ having uniform multiplicity infinity and prove that some classical Banach spaces possess this property. As an application, we show that if $\sigma\subset [0,1]$ is right closed and contains $1$, then there exists a quasinilpotent operator $T$ on a class of Banach spaces with uniform multiplicity infinity such that $\Lambda(T)=\sigma$.

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math.CA 2026-07-02

Tiling condition transfers spectrality to product measures

by Mihail N. Kolountzakis, Chun-kit Lai +2 more

Spectrality of factors of product spectral measures

When A tiles {0..N-1} by direct sum, Lebesgue on A+[0,1] times ν is spectral exactly when ν is

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We refine the method by Greenfeld and Lev for the product spectral set problem and generalize the theorem to a singular measure setting. Furthermore, we establish a new class of spectral unions of intervals for which the product spectral set question has a positive answer. More precisely, if $A$ is a subset of the natural numbers such that $A\oplus B = \{0,1,\cdots, N-1\}$ for some $B\subset \mathbb N$ and $N>1$ then the product measure $\mathcal{L}|_{A+[0,1]}\times \nu$ is a spectral measure (that may be singular) if and only if $\nu$ is a spectral measure.

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math.FA 2026-07-02

No indecomposable Banach space has the primary factorisation property

by Antonio Acuaviva, Bence Horváth +1 more

Pure infiniteness and primary factorisation

Absence for real and complex cases follows from relating the property to infiniteness of the quotient B(E)/M_E under the unique-maximal-idea

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We show that there is no real or complex indecomposable Banach space with the primary factorisation property (PFP). We relate the PFP of a Banach space $E$ to ring-theoretic infiniteness of $\mathcal{B}(E)$ and of $\mathcal{B}(E)/\mathcal{M}_E$, where $\mathcal{M}_E$ denotes the set of operators not factoring the identity on $E$, in the case it is the unique maximal ideal of $\mathcal{B}(E)$. For complex $E$ with the PFP, this quotient is purely infinite exactly when it is not scalar. We isolate the quantitative gap relevant to ultrapowers, identify classical sequence spaces as positive non-scalar cases, and show that Read's space $E_{\operatorname{R}}$ does not have the uniform PFP.

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math.FA 2026-07-02

Friedrichs realizations bijective exactly when I minus φ Q0 is invertible

by Sandeep Kumar Soni

Boundary quadruples and bijective realisations of abstract Friedrichs operators

Non-expansive parameters then automatically give m-accretive bijective realizations, with the reference operator Q0 having norm less than on

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The theory of boundary quadruples and boundary triples is well-studied for symmetric and skew-symmetric operators and in general for dual-pairs. This paper adapts a suitable version for abstract Friedrichs operators and addresses the following questions: which parameters yield bijective realisations, and which parameters yield $m-$accretive realisations. We study a boundary-quadruple framework in which closed realisations are parametrised by closed relations in a boundary space. This yields the intrinsic criterion \[T_\Theta\ \rm{is} \ \rm{bijective}\iff \lK=\Theta\dotplus \Gamma(\ker T_1) \;.\] For bounded operator parameters $\phi:\lK_1\to \lK_0$ in the boundary space, we introduce the reference operator \[Q_0=\Gamma_1(\Gamma_0|_{\ker T_1})^{-1}\,,\] prove that $\|Q_0\|< 1$, and obtain the exact criterion \[T_\phi \ \rm{is} \ \rm{bijective} \iff \I_{\lK_0}-\phi Q_0\ \rm{is} \ \rm{bijective}\;.\] Consequently, every non-expansive parameter gives a bijective realisation with signed boundary map, which is also $m-$accretive. An existence criterion for boundary quadruples and boundary triples is established in terms of (V)-boundary conditions. The multiplicity of $M-$operators associated with a fixed (V)-boundary condition is addressed in an explicit way and a parametrisation of such operators is given. The theory is illustrated by a first-order ordinary differential operator and by the stationary diffusion equation, where $Q_0$ is identified as a Cayley transform of the Dirichlet-to-Neumann operator.

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math.FA 2026-07-02

Only Jordan blocks yield circular numerical ranges among companion matrices

by Hsin-Yi Lee, Wei-Qiang Huang

On Circular Numerical Ranges of Companion Matrices with Repeated Eigenvalues

For size greater than 3 with all eigenvalues equal, circularity forces the single Jordan structure.

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We prove that if an $n\times n\ (n > 3)$ companion matrix $A$ with the spectrum $\sigma(A) = \{ a \}$ has a circular numerical range, then $A$ is the Jordan block. This problem can be described by examining zeros of the Laurent polynomial arising from geometric properties of the numerical range. The difficulty is that the relevant Laurent coefficients involve both the repeated eigenvalue $a$ and the radius parameter $\lambda$, so direct coefficient comparison does not isolate $a$. We address this by decomposing the relevant matrix into a tridiagonal Toeplitz part plus a rank-two update and using Chebyshev polynomials of the second kind. This reduction yields an explicit Laurent-coefficient formula whose vanishing under the circularity condition gives $a=0$. Furthermore, we extend this result when the spectrum is $\sigma(A) = \{0, a\}$ with algebraic multiplicities $n-m$ and $m$, respectively.

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math.PR 2026-07-02

Non-symmetric forms need distinct energy integral analysis

by Kazuhiro Kuwae, Takumu Ooi +2 more

Energy integrals and asymmetric co-potentials for closed forms

Measures of finite energy integrals and co-potentials differ from symmetric cases across three comparison views.

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We investigate the class of measures of finite energy integrals and the behavior of potentials and co-potentials associated with non-symmetric closed forms. In particular, we compare these objects with their symmetric counterparts from three viewpoints: a non-symmetric version of Stollmann--Voigt's inequality, non-symmetric perturbations of symmetric forms, and closed forms associated with non-symmetric jump-type forms. Our results indicate that measures of finite energy integrals, potentials, and co-potentials behave differently in the non-symmetric setting, requiring more delicate analysis than in the symmetric case.

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math.AP 2026-07-02

Heat energy map is C∞ smooth near identity perturbation

by Luca Di Persio, Riccardo Molinarolo

Shape analysis in Schauder spaces of the energy of heat problems in perturbed annular domains

Domain-to-energy map for Dirichlet and Neumann heat problems stays infinitely differentiable for small inner-boundary changes in an annulus.

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This paper is devoted to the shape analysis of the energy of boundary value problems for the heat equation in a bounded perforated domain $\Omega^o \setminus \overline{\Omega^i[\phi]}$ of $\mathbb{R}^n$, where the outer boundary is fixed, and the inner boundary is given by a $C^{1,\alpha}$-perturbation $\phi$ of the boundary of a reference cavity. Under standard Dirichlet or Neumann boundary conditions, we prove that, in a suitable neighborhood of the identity $\phi_0$, the domain-to-energy map is of class $C^{\infty}$. The proof is based on the construction of a global diffeomorphism, smoothly depending on $\phi$, from the reference annulus onto the perturbed one, on a decomposition of the fixed domain into regions near, intermediate to, and far from the cavity, and on the smooth dependence of the layer heat potentials upon support perturbations.

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math.OA 2026-07-02

State-space yields Davis-Wielandt shell bounds in C*-algebras

by Debarati Bhattacharya, Fuad Kittaneh +2 more

On the geometry of the algebraic Davis--Wielandt shell and norm-parallelism in C^*-algebra

Geometric properties established and radii of sums bounded, with links to norm-parallelism.

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This article is devoted to the study of the Davis--Wielandt shell and the Davis--Wielandt radii of elements in a $C^*$-algebra. Utilizing a state-space approach, several geometric properties of the algebraic Davis--Wielandt shell are established. Upper and lower bounds for the algebraic Davis--Wielandt radii are obtained including the Davis--Wielandt radius of the sum of $k$ elements. We also explore the relationship between norm-parallelism and the Davis--Wielandt radii of elements.

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math.AP 2026-07-02

Measure solutions show asynchronous exponential growth

by Christian Düll, József Z. Farkas +2 more

Asynchronous exponential growth for structured population models in measure space

Classical convergence to a one-dimensional attractor extends to Radon measures under flat metric when conditions hold.

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This paper studies the asymptotic behaviour of a structured population model on the space of nonnegative Radon measures. Such formulations naturally arise when solutions develop concentration phenomena or when the population is represented by discrete cohorts. Asynchronous exponential convergence of measure solutions towards a one-dimensional global attractor is established. While such results are classical in the $L^1$ setting, their extension to measure spaces requires different compactness and spectral arguments. We identify conditions under which the classical asymptotic behaviour persists in the space of Radon measures endowed with the flat metric, thereby extending the theory of asynchronous exponential growth beyond the classical $L^1$ framework.

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math.FA 2026-07-02

No constant bounds convex-hull representations for Weibull(r) processes

by Xuanang Hu, Hanchao Wang

Failure of Convex-Hull Bounds under Log-Convex Tails

For r in (0,1) the L_log(k+2) norms of auxiliary vectors cannot be controlled uniformly by the expected supremum, even with arbitrary choice

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Fix $0<r<1$, and let $X_1,X_2,\dots$ be independent symmetric Weibull$(r)$ random variables, that is, \[ \textsf{P}(|X_i|>t)=e^{-t^r},\qquad t\ge 0. \] We prove that there is no constant $C_r$, depending only on $r$, with the following universal property: for every finite set $T\subset \R^N$ there exists a sequence $(y_k)_{k\ge 1}\subset \R^N$ such that \[ T-T\subset conv\{y_k:k\ge 1\}, \qquad \|X_{y_k}\|_{L_{\log(k+2)}}\le C_r\,\bx(T) \quad (k\ge 1), \] where $X_t=\sum_i t_i X_i$ and $\bx(T)=\textsf{E}\sup_{t\in T}X_t$. This gives a negative answer to a question of Lata{\l}a concerning the validity of convex-hull bounds for canonical Weibull processes. In fact, the failure persists even when the auxiliary vectors appearing in the convex hull are allowed to be arbitrary.

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math.FA 2026-07-01

Finite-Hausdorff metric spaces admit relaxed distortion maps

by Roman D. Oleinik

On one relaxation of the bounded-length-distortion condition in the context of metric measure spaces

A reference measure lets the relaxed length-distortion condition hold for maps into finite-dimensional normed spaces.

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We reformulate the bounded-length-distortion condition for maps between metric spaces in a certain relaxed form that requires the presence of a reference measure on the source space, which makes the new approach more natural from the perspective of maps from metric measure spaces to metric spaces. In terms of the introduced notion, we establish some mapping results in an entirely singular setting of the following general structure: a metric measure space of finite Hausdorff dimension admits a map with the relaxed bounded-length-distortion condition into a finite-dimensional normed space.

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math.OA 2026-07-01

Graph products admit L_p-bounded Hilbert transforms

by Xiao-Qi Lu, Runlian Xia

Hilbert transforms on graph products of finite von Neumann algebras

A length-dependent Cotlar identity plus a Haagerup-type inequality extends free-product boundedness to graph products and settles an Ozawa c

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We study Hilbert transforms on graph products of finite von Neumann algebras, with particular interests on their boundedness on the associated noncommutative $L_p$-spaces for $1<p<\infty$. We establish a generalized Cotlar identity for Hilbert transforms, valid on operators whose lengths exceed a constant depending only on the underlying graph. We further prove that graph products of finite von Neumann algebras satisfying a Haagerup-type inequality admit $L_p$-bounded Hilbert transforms, therefore extending the corresponding result of Mei and Ricard for free products of finite von Neumann algebras. In addition, we obtain several equivalent characterizations of this Haagerup-type inequality and show, in particular, that it is equivalent to the graph product being generated by finite-dimensional von Neumann algebras with uniformly bounded dimensions. Our results apply, in particular, to graph products of finite groups, right-angled Hecke von Neumann algebras, and graph products of finite quantum groups. As an application, we provide positive answers to a compactness problem posed by Ozawa in the setting of graph products of finite groups and right-angled Hecke von Neumann algebras.

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math.OA 2026-07-01

Nuclearity of operator systems equals approximate factorization

by Roy Araiza, Larissa Kroell +2 more

Approximate factorization properties for operator systems

Standard conditions on the tensor product hierarchy match factorization through finite-dimensional systems.

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We show that many of the standard nuclearity properties considered in the literature for the hierarchy of operator system tensor products can be expressed as approximate factorization properties, generalizing the well-known Completely Positive Approximation Property for nuclear C*-algebras due to Choi and Effros and its generalization to nuclear operator systems due to Han and Paulsen.

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math.FA 2026-07-01

Δ₂-fundamental sequences characterize discrete Orlicz operators

by David E. Edmunds, Jan Krejčí +2 more

Operators on Orlicz sequence spaces and Delta₂-fundamentality

Boundedness of maximal and averaging operators holds exactly when the lim sup of consecutive term ratios satisfies a simple finiteness condi

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A classical result states that the Hardy--Littlewood maximal operator is bounded on an Orlicz space $L^A(\mathbb{R}^n)$ if and only if its conjugate Young function $\tilde{A}$ satisfies the $\Delta_2$-condition. The same condition also characterizes the boundedness on $L^A(0,\infty)$ of the Hardy averaging operator. We consider a discrete analogue of the problem, extended to a general interpolation framework. We offer several characterizing conditions for the boundedness of discrete maximal and average operators on Orlicz spaces. Although the principal result is as expected, for its proof some new techniques have to be developed. To this end, we introduce a new notion of the so-called $\Delta_2$-fundamental sequence, and give its interesting characterization by a simple condition involving only a limes superior of the ratio of two subsequent terms. We also prove a dual statement concerning operators of Copson type.

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math.CO 2026-07-01

Covariance bounds degree-weighted Fourier overlap for monotone functions

by Fan Chang, Hong Liu +1 more

The sharp diagonal spectral correlation inequality on the discrete cube

The inequality Cov(f,g) ≥ 4 ∑ |S| ˆf(S)² ˆg(S)² holds with equality only for disjoint supports, common dictatorships, or the AND-OR pair.

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We prove the sharp diagonal spectral correlation conjecture of Friedgut, Kahn, Kalai and Keller, proposed in their Fourier-analytic approach to Chv\'atal's conjecture. For every pair of increasing Boolean functions $f,g:\{0,1\}^n\to\{0,1\}$, $$\mathrm{Cov}(f,g)\ge4\sum_{\varnothing\ne S\subseteq[n]}|S|\hat{f}(S)^2\hat{g}(S)^2.$$ Thus covariance controls the degree-weighted collision of the two nonconstant Fourier spectra, giving a sharp Fourier strengthening of the Harris--Kleitman inequality. The theorem also implies the unweighted diagonal conjecture of Friedgut--Kahn--Kalai--Keller for an increasing family and a maximal intersecting family. The factor $4$ is optimal, and we determine all equality cases. Apart from pairs whose relevant coordinate sets are disjoint, equality occurs only for a common dictatorship and, up to relabelling coordinates and interchanging $f$ and $g$, for the two-coordinate AND-OR pair $(f,g)=(x_i x_j,\,x_i\vee x_j).$ The main novelty is a correlated four-restriction induction and a sharp endpoint convolution inequality. The usual two-restriction induction behind Harris--Kleitman sees only the parallel restricted pairs and loses the mixed Fourier information needed to control the degree-weighted diagonal spectral energy. We instead couple the four codimension-one restricted pairs with correlation $1/2$; this precise correlation extracts the missing degree-weighted energy as a nonnegative square.

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math.CA 2026-07-01

Beckmann boundary obeys Talagrand inequality on the cube

by Paata Ivanisvili, Xinyuan Xie +1 more

A Beckmann boundary form of Talagrand's conjecture on the discrete cube

New nonlocal measure is smaller than or equal to edge boundaries yet satisfies the variance times sqrt(log term) lower bound for every nonco

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We introduce the Beckmann boundary of a Boolean function \[ \mathsf{B}(f)=\inf_{\operatorname{div} V=Lf}\mathbb E\|V(x)\|_2. \] Here \[ L=\sum_iD_i,\qquad D_i f(x)=\frac{f(x)-f(x^{\oplus i})}{2}, \] and $\operatorname{div} V(x)=\sum_i (V_{i}(x)-V_{i}(x^{\oplus i}))$. This nonlocal quantity is no larger than the usual two-sided, one-sided, colored, optimized colored, or optimized fractional colored boundaries. Nevertheless, every nonconstant Boolean $f$ satisfies \[ \mathsf{B}(f)\gtrsim \operatorname{Var}(f) \sqrt{\log\!\left(1+\frac{1}{\sum_i\operatorname{Inf}_i(f)^2}\right)}. \] We also prove strong one-sided fractional spectral estimates. If $A\subset\{-1,1\}^n$ and \[ h_{A}(x)=\#\{i:x\in A,\ x^{\oplus i}\notin A\}, \] then, for $0<\alpha<1$, \[ \sum_{S\ne\varnothing}|S|^\alpha\widehat{\mathbf 1_{A}}(S)^2 \lesssim_\alpha \mathbb E\omega_\alpha(h_{A}), \] where $\omega_\alpha(m)=\sqrt m$ for $\alpha<1/2$, $\omega_{1/2}(m)=\sqrt m\log(e+m)$, and $\omega_\alpha(m)=m^\alpha$ for $\alpha>1/2$. These profiles are sharp, up to $\alpha$-dependent constants, for majority. We also show that the comparison is genuinely nonreversible: an explicit quotient-cube family makes the optimized fractional, and hence optimized colored, boundary exceed $\mathsf{B}$ by a factor $\gtrsim\sqrt{\log n}$. We further obtain a driftless Bernstein-multiplier inequality.

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math.OA 2026-07-01

II1 factors without crossed product decompositions found

by Adriana Fernández Quero, Adrian Ioana +1 more

A class of II₁ factors without non-trivial crossed product decompositions

New separable examples restrict embeddings into their tensor squares to only the canonical maps.

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We introduce a class of separable II$_1$ factors $M$ admitting no non-trivial crossed product decompositions: $M\not\cong B\rtimes_\sigma G$, for any trace preserving action $G\curvearrowright^\sigma (B,\tau)$ of an infinite countable group $G$ on a tracial von Neumann algebra $(B,\tau)$. These provide the first examples of II$_1$ factors that do not arise as crossed products of noncommutative dynamical systems. Our approach relies on a novel construction of separable II$_1$ factors $M$ whose embeddings into their tensor product square $M\overline{\otimes}M$ all arise from the canonical embeddings $x\mapsto x\otimes 1$ and $x\mapsto 1\otimes x$.

0

math.SP 2026-07-01

First-variation formula for eigenvalues holds at essential spectrum edge

by Denis Vinokurov

Eigenvalue optimization via a first-variation formula

Clarke subdifferential supplies a tool that characterizes all optimal weights for weighted Laplace and Steklov problems.

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We compute the Clarke subdifferential of the $k$th eigenvalue functional on the space of self-adjoint operators, obtaining a first-variation formula that remains valid even when the eigenvalue lies at the edge of the essential spectrum. This formula provides an effective tool for describing the structure of critical points in eigenvalue optimization problems and can also yield simple proofs of the existence of optimizers. We illustrate these advantages through applications to the optimization of weighted Laplace and Steklov eigenvalues. In particular, we characterize all optimal weights, thereby answering some open questions posed by Kokarev, and give a short proof that such weights exist.

0

math.AP 2026-07-01

Neumann Ornstein-Uhlenbeck semigroup on trees is Markovian with Gaussian invariant

by Sahiba Arora, Marjeta Kramar Fijavž +2 more

Ornstein--Uhlenbeck semigroup on rooted trees

Form methods on rooted metric trees yield a Markovian Neumann realization whose unique invariant is the Gaussian measure, plus spectral redu

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We study Ornstein--Uhlenbeck operators on rooted metric trees equipped with a Gaussian-type measure. Using form methods, we construct Dirichlet and Neumann realisations corresponding, respectively, to killing and reflection at the root. The associated semigroups are symmetric, analytic and positivity preserving; the Dirichlet semigroup is sub-Markovian, while the Neumann semigroup is Markovian and admits the Gaussian measure as its unique invariant measure up to scalar multiples. We prove compactness of the resolvent and derive linear eigenvalue asymptotics. For regular rooted trees, we adapt the Naimark--Solomyak decomposition to the Gaussian weighted setting, reducing the operators to one-dimensional half-line problems and obtaining refined spectral localisation and lower bounds.

0

math.PR 2026-07-01

Free SDEs gain global well-posedness under local Lipschitz conditions

by Jiaxin Wei, Zhi Yin

Well-posedness and stationary distribution of free stochastic differential equations

Local operator Lipschitz and Lyapunov conditions ensure unique solutions and stationary distributions in noncommutative probability spaces.

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This paper studies free stochastic differential equations driven by free Brownian motion. Under local operator Lipschitz and Lyapunov-type conditions on the coefficients, we prove the global well-posedness of solutions in the noncommutative probability setting using free It\^o calculus. We further establish the existence and uniqueness of stationary solutions under appropriate dissipativity conditions. Our results extend classical theory to the free probability framework.

0

math.FA 2026-07-01

Even or odd polynomials block Gabor frames at density N+1

by Alexander Ulanovskii, Ilya Zlotnikov

Periodic Non-uniqueness Sets for Shift-invariant Spaces and Parity-Based Obstructions to the Frame Property for Gabor Systems

An explicit algorithm finds such a low-degree window for any N when the lattice has density exactly N+1.

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The goal of this note is twofold. First, we provide explicit examples of periodic (though not necessarily lattice) sets that give rise to Gabor systems failing to form frames. Our constructions depend only on the parity of the window function $g$. Second, for a wide range of finite-dimensional function spaces $V$ we show that $V$ contains a function $g$ such that a lattice of high density fails to generate a Gabor frame. In particular, we prove that the Gr\"ochenig-Lyubarskii theorem is sharp in the finite-dimensional space of polynomials with Gaussian weight. More precisely, for every $N\in\mathbb{N}$ and every $\alpha,\beta>0$ satisfying $\alpha\beta=\frac{1}{N+1}$, we give an explicit algorithm for finding an even or odd polynomial $p$ of degree at most $N$ such that $\mathcal{G}(p(x)e^{-\pi x^2}, \alpha\mathbb{Z} \times \beta\mathbb{Z})$ does not form a frame. The proofs are constructive, elementary, and based on linear algebra.

0

math.FA 2026-07-01

Bilinear CZ operators bound L^p products on Vilenkin groups

by Adil Shafi Wani, Qaiser Jahan +1 more

Bilinear Calder\'{o}n-Zygmund operators on Vilenkin groups

They map L^{p1} × L^{p2} to L^p when 1/p equals the sum of the reciprocals and extend similarly to Morrey spaces.

abstract click to expand

In this article, we study bilinear Calder\'on--Zygmund operators on a Vilenkin group $G$. As a preliminary step, we establish a Grafakos--Torres-type endpoint weak-type result in our setting. Furthermore, we prove that such operators extend to bounded bilinear mappings from $L^{p_1}(G)\times L^{p_2}(G)$ into $L^p(G)$ under the natural condition $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}.$ We then obtain a corresponding boundedness result in Morrey spaces, showing that these operators extend to bounded bilinear mappings from $\mathcal{M}_{p_1,u_1}(G)\times \mathcal{M}_{p_2,u_2}(G)$ into $\mathcal{M}_{p,u}(G)$ under suitable assumptions. These results generalize the classical bilinear estimates to the setting of Vilenkin groups.

0

math.FA 2026-07-01

Gabor bases on local fields minimize ambiguity function support

by Kumar Abhinav, Qaiser Jahan

Gabor Orthonormal Bases with Maximal Localization and Gabor Frame Operator on Local Fields

Explicit construction achieves maximal localization impossible on the real line due to uncertainty.

abstract click to expand

We provide an explicit construction of a Gabor orthonormal bases for a local field $K$ that provides maximal localization in both time and frequency. Such a localization is not true in case of $\mathbb{R}$ due to the uncertainty principle. In particular, we construct examples of functions $f \in L^2(K)$ such that the support of the ambiguity function of $f$ is of minimum measure. Moreover, we establish a quantitative uncertainty principle for local fields, which follows as a consequence of Lieb's inequalities for general locally compact abelian group. In addition, we develop fundamental operator representations for Gabor systems defined over local fields.

0

math.FA 2026-07-01

Homogeneous order maps on C0 cones are weighted compositions

by Natsumi Shibata, Izuho Matsuzaki +1 more

Order Isomorphisms between Positive Cones of C₀(X)

They arise from homeomorphisms between the spaces and continuous weights, and extend to linear isomorphisms on the full function spaces.

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Let $X$ and $Y$ be locally compact Hausdorff spaces. We study order isomorphisms \[ T:C_0^+(X)\to C_0^+(Y), \] where $C_0(X)$ denotes the Banach space of all real-valued continuous functions on $X$ vanishing at infinity, and \[ C_0^+(X)=\{f\in C_0(X):f\ge0\} \] is its positive cone. We assume that $T$ is positive homogeneous. That is, \[ T(rf)=rT(f) \qquad (r>0,\,f\in C_0^+(X)). \] Under this assumption, we prove that $T$ is represented as a weighted composition operator induced by a homeomorphism from $Y$ onto $X$ and a bounded continuous weight function. Moreover, we show that $T$ extends uniquely to a linear order isomorphism between $C_0(X)$ and $C_0(Y)$.

0

math.FA 2026-07-01

Segal algebra S0 defines stochastic processes on groups

by Hans G. Feichtinger, Wolfgang Hörmann

A Distributional Approach to Generalized Stochastic Processes on Locally Compact Abelian Groups

A functional-analytic method using only S0(G) avoids vector-valued integration for processes on locally compact abelian groups.

abstract click to expand

This paper is dedicated to Paul Butzer on the occasion of his 85th birthday. His work and example have strongly influenced not only the first author, but also generations of mathematicians working in approximation theory and Fourier analysis. He has shown younger colleagues the importance of remaining open to applied areas, avoiding an overly narrow scope, and exploring different ways of understanding mathematical facts. A recurring theme in his work is the logical equivalence of fundamental statements in analysis. It may be less widely known that, besides his central role in approximation theory, Paul Butzer has also made significant contributions to probability theory. We hope that he will enjoy this note, which shows that a purely functional-analytic treatment of generalized stochastic processes is possible. The approach is based on the Segal algebra S0(G) and avoids several technical difficulties associated with the customary framework of vector-valued integration and topological vector spaces.

0

math.AP 2026-07-01

Comparison principle for unbounded viscosity solutions on Wasserstein spaces

by Charles Bertucci (CEREMADE), Pierre-Louis Lions (CdF (institution) +1 more

Non-linear Stegall's lemma and general Hamilton-Jacobi-Bellman equations on Wasserstein spaces

Extending Stegall's lemma yields uniqueness for Hamilton-Jacobi-Bellman equations whose state is a probability measure.

abstract click to expand

We present a comparison principle for unbounded viscosity solutions to Hamilton-Jacobi equations on Wasserstein spaces of probability measures over $R^d$ . In addition to the use of standard techniques of viscosity solutions, our approach requires a key extension on Wasserstein spaces of a result of perturbed optimization on Banach spaces due to Stegall.

0

math.AP 2026-07-01

Dual minimax formulas give real levels for non-selfadjoint pencils

by Yavdat Il'yasov, Nur Valeev

Cone Minimax Principles for Non-Selfadjoint Operator Pencils

Sup-inf and inf-sup principles on admissible cone pairs match principal spectral values even when the weight operator is singular.

abstract click to expand

We propose a variational approach to principal spectral values of non-selfadjoint operator pencils $\mathcal L u=\lambda\mathcal G u$, where the weight operator $\mathcal G$ may be singular. The aim is to obtain Rayleigh-type minimax formulas for selected real spectral levels in settings where the standard selfadjoint variational theory is unavailable and positivity-based methods may not apply directly. The construction is based on the extended two-variable Rayleigh quotient \[ \mathcal R(u,v) = \frac{\langle \mathcal L u,v\rangle} {(\mathcal G u,v)_H},\] defined on admissible cone pairs. It leads to dual sup-inf and inf-sup principal levels, cone quasi-eigenvalues, and corresponding trapping and saddle-point principles. The resulting minimax formulas characterize selected real cone levels of non-selfadjoint operator pencils and identify them with principal spectral values whenever positive right-left eigenpairs exist, including cases with non-invertible operators and singular weights. We prove that these formulas are stable under finite-dimensional approximation. Thus the classical idea of approximating spectral data by finite-dimensional variational problems acquires an analogue for non-selfadjoint operator pencils in an ordered cone setting. The method also yields a posteriori spectral certificates, one-sided perturbation bounds, and approximation estimates. Elliptic examples illustrate both the scope of the method and the sharpness of the estimates.

0

math.CV 2026-07-01

Pluriharmonic functions on polydisc obey explicit product integral bounds

by Suman Das, Antti Rasila +1 more

Isoperimetric-type inequalities for pluriharmonic functions on the polydisc

The inequality uses cosine constants from one-variable Riesz estimates and controls weighted Bergman norms by Hardy norms.

abstract click to expand

We prove isoperimetric-type inequalities for pluriharmonic functions in the unit polydisc $\mathbb{U}^n$. Let $h^p(\mathbb{U}^n)$ and $b^p_{\mathbf{q}}(\mathbb{U}^n)$ denote, respectively, the pluriharmonic Hardy space and the pluriharmonic weighted Bergman space in $\mathbb{U}^n$. We prove that if $m\in\mathbb{N}$, $m\geq2$, $1<p_1,\ldots,p_m<\infty$, and $f_j\in h^{p_j}(\mathbb{U}^n)$, then \[ \int_{\mathbb{U}^n}\prod_{j=1}^m |f_j(z)|^{p_j}\,d\mu_{\mathbf{m-2}}(z) \leq \prod_{j=1}^m \left[ \frac{\sqrt2\cos\left(\frac{\pi}{2mp_j}\right)} {\sqrt{1-|\cos(\pi/p_j)|}} \right]^{p_j} \prod_{j=1}^m \|f_j\|_{h^{p_j}(\mathbb{U}^n)}^{p_j}. \] In particular, \[ \|f\|_{b^{mp}_{\mathbf{m-2}}(\mathbb{U}^n)} \leq \frac{\sqrt2\cos\left(\frac{\pi}{2mp}\right)} {\sqrt{1-|\cos(\pi/p)|}} \|f\|_{h^p(\mathbb{U}^n)}. \] We also prove the following inclusion theorem: If $f\in h^2(\mathbb{U}^n)$, then \[ \|f\|_{h^{2n}(\mathbb{B}_n)} \leq \sqrt2\cos\left(\frac{\pi}{4n}\right) \|f\|_{h^2(\mathbb{U}^n)}, \] where $\mathbb{B}_n$ is the unit ball in $\mathbb{C}^n$. A corresponding ball-volume inequality is obtained as well. The constants are explicit and are obtained from sharp Riesz-type estimates. In the planar case, they coincide with the best available constants in the literature, although sharpness of the resulting pluriharmonic inclusions remains open.

0

math.FA 2026-06-30

Fractional operators bounded on generalized Fofana spaces

by Pokou Nagacy, Bérenger Akon Kpata +1 more

Fractional integral and fractional maximal operators on generalized Fofana spaces

Boundedness extends Morrey-type theory and produces new inequalities relating the Riesz potential to generalized fractional integrals.

abstract click to expand

Generalized Fofana spaces were recently introduced as generalizations of Fofana spaces and Nakai's generalized Morrey spaces. In this paper, we establish the boundedness properties of the following operators in these spaces: fractional integral operators, fractional maximal operators and generalized fractional integral operators. As a consequence, we obtain generalized Olsen-type inequalities involving the Riesz potential and generalized fractional integral operators.

0

math.FA 2026-06-30

Surjective isometries on positive unit sphere extend linearly

by Yuta Enami, Daisuke Hirota +2 more

Surjective isometries on the positive parts of the unit spheres of some function spaces

In C1 and Lip spaces with derivative-weighted norms, maps on positives extend to complex-linear isometries on the whole space.

abstract click to expand

We consider the space $C^1[0, 1]$ of continuously differentiable functions on the closed unit interval $[0, 1]$ and the space $\operatorname{Lip}[0, 1]$ of Lipschitz continuous functions on $[0, 1]$, equipped with the norms \begin{align*} \|f\|_{\sigma, p} = \begin{cases} \sqrt[p]{|f(0)|^p + \|f'\|_\infty^p} & (1 \le p < \infty), \\ \max\{\, |f(0)|, \|f'\|_\infty \,\} & (p = \infty). \end{cases} \end{align*} We show that every surjective isometry on the positive part of the unit sphere extends to a surjective complex-linear isometry on the entire space. As a corollary, every such isometry also extends to an isometric order isomorphism on the real subspaces $C^1_{\mathbb{R}}[0, 1]$ and $\operatorname{Lip}_{\mathbb{R}}[0, 1]$.

0

math.FA 2026-06-30

Banach space pairs show arbitrarily large embeddability ratios

by Avik Das

Coarse Embeddability Ratios of Banach Spaces

The invariant CR(X, E) grows without bound for some pairs, showing no universal limit exists on this embeddability measure.

abstract click to expand

Given two Banach spaces $X$ and $E$, one can associate a numerical invariant $\mathcal{CR}(X, E)$, called the coarse embeddability ratio, which provides a criterion for coarse and uniform embeddability. We compute the coarse embeddability ratio for several important classes of Banach spaces, using various tools from the nonlinear theory of Banach spaces. Finally, we find pairs of Banach spaces with arbitrarily large coarse embeddability ratio, resolving an open problem of Rosendal in the negative.

0

math.LO 2026-06-30

ReLU nets approximate traceable sets at rate O(ε^{-p(n-1)/m})

by Clemens Kinn, Philipp Petersen

Fast approximation and learning of binary classification tasks in o-minimal structures using ReLU neural networks

Rates independent of ε for depth yield classification error decaying as N^{-m/(m+pn-p)} on uniform samples.

abstract click to expand

We study binary classification problems whose decision sets are given by definable sets in o-minimal expansions of the real field. Motivated by cell decomposition of definable sets, we introduce traceable sets as a classical proxy for definable decision regions and analyze their approximation by ReLU neural networks. Under uniform bounds on the number of connected components and suitable $C^m$ extensions for the boundary functions, we prove that characteristic functions of traceable subsets of $[-1/2,1/2]^n$ can be approximated in $L^p$ to accuracy $\varepsilon>0$ by ReLU neural networks of size $\mathcal{O}(\varepsilon^{-p(n-1)/m})$, with depth independent of $\varepsilon$ and polynomially bounded weights. This establishes quantitative approximation rates for certain definable collections in o-minimal structures using ReLU neural networks. The same approach also yields the stated approximation rates for a subclass of definable maps $[-1/2,1/2]^n \to \mathbb{R}$. We then combine the approximation capabilities with entropy estimates for ReLU neural network classes to obtain statistical learning rates for empirical risk minimization with hinge loss. For $N$ uniformly distributed samples, the resulting classifiers achieve expected misclassification error of order $N^{-m/(m+pn-p)}$ up to an arbitrarily small polynomial loss.

0

math.FA 2026-06-30

Mercer expansions converge uniformly on Sobolev spaces without positive definiteness

by Daniel Constantin Rademacher

Sobolev-Mercer Expansions and Applications to Stochastic Processes

When order k exceeds dimension, the expansions apply to covariance kernels and approximate both random fields and their derivatives.

abstract click to expand

We establish a fundamental extension of Mercer's celebrated theorem by introducing a class of higher-order kernel operators acting on Sobolev spaces $H^k(\Theta)$, where $\Theta \subset \mathbb{R}^d$ is a bounded domain and $k\in\mathbb{N}_0$ corresponds to the order of weak differentiability. The spectral decomposition of these operators then yields Mercer-type expansions that are optimal in $H^k(\Theta\times\Theta)$. Notably, we derive from the embedding properties of Sobolev spaces, that for $k>d$, these expansions also converge uniformly without requiring the kernel to be positive definite. For positive definite kernels, we confirm the nuclearity of these higher-order operators and establish a significant refinement of Mercer's Theorem. These results lead to novel spectral representations of RKHS and have subtle implications for stochastic analysis. Applied to the covariance kernels of weakly differentiable random fields, our theory provides refined Karhunen-Loeve expansions that facilitate the simultaneous mean-square optimal approximation of both the process and its derivatives.

0

math.FA 2026-06-30

Small-2 sets coincide with Riesz sets

by A. To-Ming Lau, A. Ülger

Small-2 Sets Are Riesz Sets

The product condition on measures forces equality with L1 and determines Arens regularity of the corresponding ideals.

abstract click to expand

Let $ G $ be a compact metrizable Abelian group, $ L^{1}(G) $ its group algebra and $ M(G) $ its measure algebra. For each proper subset $ E $ of the dual group $ \hat{G} $, let $ L^{1}_{E}(G)=\{f\in L^{1}(G):\hat{f}=0 \text{ on } \hat{G}\setminus E \}$ and $M_{E}=\{\mu\in M(G):\hat{\mu}=0 \text{ on }\hat{G}\setminus E\} $. If $ M_{E}(G)=L^{1}_{E}(G) $ then the set $ E $ is said to be a Riesz sets. If $ M_{E}(G)*M_{E}(G)\subseteq L_{E}^{1}(G) $ then $ E $ is said to be a small-2 set. The main results of this paper are the following: 1. Every small-2 set is a Riesz set. 2. The ideal $ L^{1}_{E}(G) $ is Arens regular iff $ E $ is a Riesz set. Let $ A=L_{E}(G) $ and equip $ A^{**} $ with the first Arens product. 3. The centre of $ A^{**} $ is $ Z(A^{**})=A+N(A^{**}) $, where $ N(A^{**})=\{r\in A^{**}:rA^{**}=\{0\}\} $. These results settle three long-standing open problems in this area.

0

math.DS 2026-06-30

Stable invariant measures exist via stochastic integrals

by Valentin Gillet

Stable invariant measures in linear dynamics

For operators and semigroups on Banach spaces with dense bilateral backward orbits or rich eigenvectors.

abstract click to expand

We study the existence of stable invariant measures for operators and strongly continuous semigroups of operators on Banach spaces admitting either a dense bilateral backward orbit or a sufficiently rich family of eigenvectors. These invariant measures are realized as the distributions of stochastic integrals with respect to stable random measures. We also discuss invariant measures with other classes of distributions for such operators and semigroups.

0

math.OA 2026-06-30

Noncommutative Wiener-Wintner theorem for amenable groups

by Panchugopal Bikram, Sudipta Kundu +1 more

Non-Commutative Wiener-Wintner theorem for amenable group actions

Decomposition into almost periodic and weakly mixing parts yields the result on finite von Neumann algebras.

abstract click to expand

Let $G$ be a locally compact second countable amenable group acting on a finite von Neumann algebra $(\mathcal{M},\tau)$ by trace-preserving automorphisms. In this article, we establish a Jacobs-de Leeuw-Glicksberg decomposition for this action, obtaining a decomposition of $\mathcal{M}$ into its almost periodic and weakly mixing components. As an application, we prove a noncommutative Wiener--Wintner theorem for amenable group actions on finite von Neumann algebras.

0

math.FA 2026-06-30

Uncertainty principle holds for locally compact abelian groups

by Hartmut Führ

Heisenberg uncertainty inequalities for locally compact abelian groups

A version is proved with lower bounds compared to the Euclidean case and formulated for local fields.

abstract click to expand

We prove a version of Heisenberg's uncertainty principle for a rather general class of locally compact abelian groups. We compare the lower bound provided by our approach with the optimal lower bound in the Euclidean case, and formulate the Heisenberg uncertainty principle for local fields.

0

math.FA 2026-06-30

Polydisc operator compactness checked only on boundary

by Anne Dorval (LMBP)

Compactness of composition operator on weighted Bergman spaces of the polydisc

Criterion uses distinguished boundary; geometric tests work for beta > d-3

abstract click to expand

We study composition operators induced by a smooth symbol between weighted Bergman spaces of the polydisc. We first prove a compactness criterion that only requires knowing what happens on the distinguished boundary. Then we prove simple geometric characterizations of boundedness and compactness on some $A^2_\beta(\mathbb{D}^d)$, particularly for $\beta > d-3$.

0

math.FA 2026-06-30

Square-root metric on circle functions is bi-invariant and complete

by Gangsong Leng, Lecheng Yang

A square-root complex inequality and its induced metric structure

It induces the L2 topology; on tori the exponent 1/2 is optimal with explicit geodesics and dimension n+1

abstract click to expand

Let $(\Omega,\mu)$ be a finite measure space with $M=\mu(\Omega)>0$. We investigate the integral form, stability, and metric geometry associated with a square-root complex. After proving the inequality and determining all equality cases, we analyze its phase stability near the intersection of the two branches of the equality set. In general phase directions, the quadratic term is precisely a Cauchy--Schwarz deficit; along the corresponding degenerate cone, the leading term is of fourth order and is strictly positive. A symmetric two-point example shows that the exponent four is unavoidable in any uniform distance-stability estimate. Finally, on the group of measurable circle-valued functions, we introduce the LY-metric \[ d_\mu(f,g)=\left|M-\int_\Omega f\overline g\,d\mu\right|^{1/2}. \] We prove that this metric is bi-invariant and complete, and that it induces the same topology as the $L^2$ metric. On finite-dimensional tori, we establish the optimality of the exponent $1/2$, derive explicit formulas for the intrinsic distance and geodesics, describe the anisotropic geometry and volume growth of small metric balls, and show that the Hausdorff dimension is $n+1$.

0

math.OA 2026-06-30

Heat semigroup on free orthogonal quantum groups hypercontracts at optimal time

by Haonan Zhang

Sharp hypercontractivity for free orthogonal quantum groups of Kac type

Result gives sharp contraction rate to equilibrium for O_F^+ of Kac type in noncommutative L^p spaces.

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We prove that the heat semigroup on the free orthogonal quantum group $O_F^+$ of Kac type satisfies hypercontractivity with the optimal time.

0

math.FA 2026-06-29

Conditions identify when Banach algebra operators have dense periodic elements

by Stefan Ivkovic

Periodicity in Banach algebras

The criteria characterize generalized weighted shifts on Hilbert modules over compact and commutative C*-algebras.

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In this paper, we consider operators that are compositions of an isometric isomorphism and a left multiplier on a Banach algebra, and we provide necessary and sufficient conditions for these operators to have a dense set of periodic elements. As an application of this result, we characterize generalized weighted shifts with a dense set of periodic elements on the standard Hilbert module over C*-algebra of compact operators on a separable Hilbert space. As another application, we characterize generalized weighted shifts with a dense set of periodic elements on the standard Hilbert module over commutative non-unital C*-algebra.

0

math.FA 2026-06-29

Besov trace space equals L_p on Ahlfors-David regular sets

by Aleksei Y. Chikalov

Traces of Besov spaces to regular subsets of metric measure spaces: the limiting case

In spaces with doubling measures and Poincaré inequalities, traces of B^{θ/p}_{p,1} coincide with L_p using the θ-Hausdorff measure on the s

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Let $(X,d,\mu)$ be a metric measure space whose measure $\mu$ is uniformly locally doubling and which supports a local weak $(1,p)$-Poincar\'e inequality for some $p\in[1,\infty)$. Given $\theta\in(0,p)$ and an Ahlfors--David codimension-$\theta$ regular subset $E\subset X$, we provide a complete intrinsic description of the trace-space of the Besov space $B^{\theta/p}_{p,1}(X)$ to $E$. More precisely, we show that the trace operator is well defined and bounded from $B^{\theta/p}_{p,1}(X)$ to $L_p(E,\mathcal H_\theta\lfloor_E)$. We also show that the upper estimate in the Ahlfors--David codimension-$\theta$ regularity condition is necessary for such boundedness under the local weak Poincar\'e inequality. Conversely, assuming that $E$ is Ahlfors--David codimension-$\theta$ regular, we construct a bounded nonlinear extension operator from $L_p(E,\mathcal H_\theta\lfloor_E)$ to $B^{\theta/p}_{p,1}(X)$. Thus the trace-space is identified intrinsically with $L_p(E,\mathcal H_\theta\lfloor_E)$. This extends the classical limiting case of the trace theorem obtained by Burenkov and Gol'dman. Finally, we apply the general theory to $K$-regular trees, $K\ge 1$, for which we additionally derive a necessary and sufficient criterion for the existence of traces.

0

math.FA 2026-06-29

Traces of weighted Besov spaces characterized on regular sets

by Aleksei Y. Chikalov

Traces of weighted Besov spaces to Ahlfors-David regular sets: the limiting case

Weakened regularity on Ahlfors-David sets plus local A_p weights yield complete trace descriptions, including for power weights on hyperplan

abstract click to expand

Given $n\in \mathbb{N}$, $p\in [1,\infty)$, and a weight $\gamma$ satisfying the local Muckenhoupt $A_p$ condition, we introduce a weakened version of the Ahlfors--David codimension-$\theta$ regularity condition for Ahlfors--David $d$-regular sets $E\subset\mathbb{R}^n$, where $d\in(0,n)$ and $\theta\in(0,p)$. Under this assumption, we provide a complete intrinsic description of the trace-space of the weighted Besov space $B^{\frac{\theta}{p}}_{p,1}(\mathbb{R}^n,\gamma)$ to $E$. In particular, our results cover the case of power-type weights $\gamma(x)=|x|^\alpha$ with $-n<\alpha<n(p-1)$, $\alpha\neq -(n-1)$, when $E=\mathbb{R}^{n-1}$. This extends earlier results obtained by Haroske and Schmeisser.

0

math.FA 2026-06-29

Circularity equals strong circularity for direct sums of irreducibles

by Soumitra Ghara, Surjit Kumar +1 more

Circular operators and their strong circularity

The same holds for every operator in the Cowen-Douglas class, restricting where counterexamples to Gellar's conjecture can appear.

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Circular operators have been studied extensively since the work of R. Gellar, who conjectured that every circular operator on a complex separable Hilbert space is strongly circular. In this short note, we show that circularity and strong circularity coincide for bounded operators that are finite or countably infinite direct sums of irreducible operators. This considerably narrows the search for potential counterexamples to Gellar's conjecture. As an application, we prove that every circular operator in the Cowen-Douglas class is strongly circular. In addition, we obtain several general results on circular operators that reveal the significance of the hyper-range and the Cauchy dual.

0

math.FA 2026-06-29

Wiener amalgam spaces prove Sobolev embeddings with general local norms

by Hans G. Feichtinger

The Concept of Wiener Amalgam Spaces

Extending local components also identifies multipliers from the Wiener algebra to its dual as mild distributions.

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This article concerns Wiener amalgam spaces % , recalls their basic properties and provides some hints about their usefulness in various branches of Harmonic Analysis. Despite the fact that the underlying construction principles % of Wiener amalgam spaces is are quite easy to understand and basic facts follow naturally by simple rules, these spaces have not obtained the same popularity as certain other function spaces which are much more complicated to describe and often just serve a very particular purpose. \newline \indent This situation has motivated the author to provide here a summary of the foundations of the theory of Wiener amalgam spaces (and the motivation behind their construction) and a selection of relevant applications, some 45 years years after the key paper published in 1983. \newline \indent We recall first that the so-called {\it classical Wiener amalgam spaces} using local $\HFLpsp$-norms combined with a global $\HFlqsp$-behaviour are already quite useful, e.g.\ for an improvement of the Hausdorff-Young Theorem with some interesting consequences for Sobolev algebras. However, the main emphasis will be based on the idea of allowing more general local components (describing for example smoothness or membership in the Fourier algebra). This opened the door to the introduction of {\it modulation spaces}, which are now recognized as standard tools in time-frequency analysis. \newline \indent We will demonstrate in this article how Wiener amalgam spaces methods can be used to prove the Sobolev embedding theorem or determine the pointwise multipliers of Sobolev algebras. We also demonstrate that the space of multipliers from the classical Wiener algebra $\HFWCOliRd$ into its dual can be identified with $\HFSOPRd$, the space of mild distributions. }

0

math.FA 2026-06-29

Equivalence classes of Cauchy sequences equal S0'

by Hans G. Feichtinger

A Sequential Approach to Mild Distributions

Sequences of bounded continuous functions recover the dual of the Segal algebra S0 without integration theory.

abstract click to expand

We describe an elementary sequential realization of the Banach Gelfand triple (S0(R^d), L2(R^d), S0'(R^d)). Here S0(R^d) is a Segal algebra of test functions, L2(R^d) is the usual Hilbert space, and S0'(R^d) is its dual space of mild distributions. This framework is fundamental for Gabor analysis and provides a natural setting for the generalized Fourier transform and the short-time Fourier transform. Inspired by Lighthill's sequential approach to tempered distributions, we construct an extended domain for the short-time Fourier transform from equivalence classes of extended mild Cauchy sequences, abbreviated as ECmiCS. Their representatives are sequences of bounded continuous functions. The construction avoids Lebesgue integration and the theory of tempered distributions. Our main result identifies the resulting sequential space canonically with S0'(R^d), thereby recovering the Banach Gelfand triple in an elementary form.

0

math.FA 2026-06-29

Convolution equips homogeneous Banach spaces as essential L1 modules

by Hans G. Feichtinger

Homogeneous Banach spaces as Banach convolution modules over M(G)

Partitions of unity define the action on groups without Haar measure, so approximate identities converge strongly to the identity.

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We develop an elementary approach to convolution and Fourier analysis on a locally compact Abelian group (G), based on bounded measures and bounded uniform partitions of unity. In earlier work, the author introduced convolution and the Fourier--Stieltjes transform on the Banach space (M(G)) of bounded measures, viewed as linear functionals, in a direct Euclidean setting. The present paper constructs arbitrarily fine bounded uniform partitions of unity on general locally compact Abelian groups. The construction is designed to avoid structure theory and does not presuppose Haar measure or Lebesgue integration. It is then used to establish a natural convolution-module structure of (M(G)) on a broad class of homogeneous Banach spaces on (G). This class includes (L^p(G)), for (1\leq p<\infty), the Fourier--Stieltjes algebra, and, in particular, Segal algebras. After introducing Haar measure, we identify (L^1(G)) with the closure in (M(G)) of the measure-embedded space (C_c(G)). We prove that the homogeneous Banach spaces under consideration are essential (L^1(G))-modules. Consequently, standard approximate identities act in the expected manner and converge strongly to the identity operator. The method follows the spirit of Hans Reiter: it avoids the customary reliance on LCA-group structure theory and on vector-valued integration arguments based on duality. It is intended as a foundation for a subsequent elementary treatment of the extended Fourier transform in the Banach Gelfand triple generated by a Segal algebra.

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math.FA 2026-06-29

Polar decomposition refines Zhang matrix-sum bound

by Jean-Christophe Bourin, Eun-Young Lee

Some hybrid matrix triangle inequalities

The decomposition of the quadratic symmetric modulus yields an operator inequality with factor √2/2 instead of the global norm bound.

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A recent result due to Teng Zhang compares the sum of $m$ matrices and the sum of their quadratic symmetric moduli: $$ \left\| \sum_{k=1}^m A_k\right\| \le \sqrt{2} \left\| \sum_{k=1}^m |A_k|_{\qsym}\right\| $$ for every unitarily invariant norm. Here $|A|_{\qsym}$ is the quadratic mean of $|A|$ and $|A^*|$. We derive operator and eigenvalue refinements of Zhang's inequality from a new polar decomposition for the quadratic symmetric modulus. For instance, $$ \left| \sum_{k=1}^m A_k\right| \le \frac{\sqrt{2}}{2} \left\{ \sum_{k=1}^m \left(|A_k|_{\qsym}+V|A_k|_{\qsym}V^*\right)\right\} $$ for some unitary matrix $V$. We also establish the polar decomposition for the maximal modulus associated with Olson's order, and derive, as in the quadratic case, a series of estimates.

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math.FA 2026-06-29

Functions with exact graph dimensions form strongly c-algebrable sets

by Jia Liu, Saisai Shi

On strong algebrability and spaceability of continuous functions and fractal dimensions

When 1 < s < r < t ≤ 2 the intersection of sets with fixed Hausdorff, lower-box and upper-box dimensions contains a continuum algebra and a

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In this paper, we investigate the strong algebrability and $(\alpha,\beta)$-lineability/spaceability of continuous functions with prescribed fractal dimensions. For $1< s< r< t\leq2$, we define $$H_s[0,1]=\{f\in C[0,1]:{\dim}_HG_f([0,1])=s\},$$ $$\underline{B}_r[0,1]=\{f\in C[0,1]:\underline{{\dim}}_BG_f([0,1])=r\}$$ and $$\overline{B}_t[0,1]=\{f\in C[0,1]:\overline{{\dim}}_BG_f([0,1])=t\}.$$ We prove that $H_s[0,1]\cap\underline{B}_r[0,1]\cap\overline{B}_t[0,1]$ is both strongly $\mathfrak{c}$-algebrable and spaceable. This complements recent findings of Bonilla et al. \cite{BFBS}, Esser et al. \cite{EMVVS}, and Liu et al. \cite{LZS}. We prove that for any $1<s\leq t\leq2$, $H_s[0,1]\cap\overline{B}_t[0,1]$ is $(p,\mathfrak{c})$-spaceable for $p=1,2$. We also prove that $H_s[0,1]\cap\overline{B}_t[0,1]$ is $(n,m+n)$-lineable for any $m,n\in\mathbb{N}$, thus complementing the recent work of Liu et al. \cite{LS}.

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math.FA 2026-06-29

Bijections on C*-algebra cones satisfying Fischer-Muszély equation extend to Jordan isomor

by Daisuke Hirota, Jyamira Oppekepenguin

On the Fischer-Musz\'ely equation for the positive cones of C^*-algebras

Maps on positive semidefinite cones extend via Jordan *-isomorphism and positive multiplier on both sides.

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We study the Fischer-Musz\'ely functional equation for the positive semidefinite and the positive definite cones of unital $C^*$-algebras. We show that any bijection between the positive semidefinite cones satisfying the Fischer-Musz\'ely equality extends to a Jordan $*$-isomorphism followed by multiplication on both sides by a positive element. As a corollary, we obtain a similar result for the positive definite cones of unital $C^*$-algebras.

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math.FA 2026-06-29

No nontrivial semigroup yields strong continuity on all of BMOA

by Austin Anderson, Mirjana Jovovic +1 more

Composition Semigroups on BMOA and H^(infty)

The maximal continuity space stays strictly smaller than BMOA, with a uniform H^∞-norm limit for every semigroup.

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We study $[\phi_t , X]$, the maximal space of strong continuity for a semigroup of composition operators induced by a semigroup $\{\phi_t\}_{t\ge0}$ of analytic self-maps of the unit disk, when $X$ is BMOA, $H^\infty$ or the disk algebra. In particular, we show that $[\phi_t,\text{BMOA}] \neq \text{BMOA}$ for all nontrivial semigroups. We also prove, for every semigroup $\{\phi_t\}_{t\ge0}$, that $\lim_{t \to 0^+} \phi_t(z) = z$ not just pointwise, but in $H^{\infty}$ norm. This provides a unified proof of known results about $[\phi_t , X]$ when $X \in \{H^p, A^p, \mathcal B_0, \text{VMOA}\}$.

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math.CV 2026-06-29

Integral operator S_g bounded below on Bloch space for specific g

by Austin Anderson

Some Closed Range Integral Operators On Spaces of Analytic Functions

Characterization given for g making the operator have closed range on Bloch, Hardy, and Bergman spaces; companion operator fails on most but

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Our main result is a characterization of $g$ for which the operator $S_g(f)(z) = \int_0^z f'(w)g(w)\, dw$ is bounded below on the Bloch space. We point out analogous results for the Hardy space $H^2$ and the Bergman spaces $A^p$ for $1 \leq p < \infty$. We also show the companion operator $T_g(f)(z) = \int_0^z f(w)g'(w) \, dw$ is never bounded below on $H^2$, Bloch, nor BMOA, but may be bounded below on $A^p$.

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math.FA 2026-06-29

Kato-Rosenblum theorem extends to unbounded n-tuples

by Rhishab Bhutani, Dan Virgil Voiculescu

A Sharp Kato-Rosenblum Type Theorem for Unbounded n-Tuples

Commuting self-adjoint operators whose difference lies in the Lorentz (n,1) ideal share the same absolutely continuous spectrum when n is at

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We prove a generalization for commuting $n$-tuples of unbounded self-adjoint operators and the Lorentz $(n,1)$ ideal,$n \ge 3$, of the Kato-Rosenblum theorem. The result is derived from earlier work for bounded operators [8]. Also, a very weak result for $n=2$ unbounded operators and other additional results are obtained.

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math.FA 2026-06-29

Maximal symbol subalgebra forms complete locally convex algebra

by Miguel Angel Rodriguez Rodriguez

Commutative topological algebras on translation-invariant reproducing kernel Hilbert spaces

Transporting from symbols to operators on translation-invariant kernel spaces produces commutative topological algebras of operators and ker

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We study commutative topological algebras naturally associated with translation-invariant reproducing kernel Hilbert spaces whose direct integral decomposition has one-dimensional fibers. Starting from the bounded algebra of translation-invariant operators, we pass to a common dense domain generated by reproducing kernels and identify the corresponding diagonalizable operators with multiplication by symbols in an intersection of weighted $L^2$-spaces. On the symbol side this gives a canonical space $\mathcal F_0$ and a maximal multiplicative subalgebra $\mathcal F_M$, which is a complete locally convex $*$-algebra. Transporting the structure back yields corresponding algebras of operators and integral kernels. We also discuss when the inclusions $L^\infty(\Omega)=\mathcal F_\infty\subset \mathcal F_M\subset \mathcal F_0$ are strict, and illustrate the results with vertical and radial operators on classical Bergman and Fock spaces.

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math.FA 2026-06-29

H∞ calculus for commuting Ritt tuples equals R-Ritt plus dilation

by Christian Le Merdy, M. N. Reshmi

Connecting H^infty-functional calculus and isometric dilations for commuting families of Ritt_E operators

Equivalence holds on UMD spaces precisely when each operator is R-Ritt_E and the family admits a polynomially bounded isometric dilation on

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Let $(T_1,\ldots,T_d)$ be a commuting $d$-tuple of Ritt$_E$ operators on some UMD Banach space $X$. We show that $(T_1,\ldots,T_d)$ admits a bounded $H^\infty$-functional calculus if and only if $T_k$ is an $R$-Ritt$_E$ operator for every $k=1,\ldots,d$, and the $d$-tuple $(T_1,\ldots,T_d)$ admits an isometric dilation $(U_1,\ldots,U_d)$ on some UMD Banach space $Y$ such that $(U_1,\ldots,U_d)$ is polynomially bounded. In the case where $X$ further possesses property $(\alpha)$, we establish other characterizations of the $H^\infty$-functional calculus property for $(T_1,\ldots,T_d)$ in terms of isometric dilations.

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math.MG 2026-06-29

Entropy convexity at Wasserstein barycenters forces Hilbertian norms

by Bang-Xian Han, Deng-Yu Liu

Wasserstein Barycenter Convexity Detects Hilbertian Geometry

In finite dimensions the inequality holds for arbitrary finite measures only when the norm comes from an inner product, unlike curvature con

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We prove that convexity of the Boltzmann entropy at Wasserstein barycenters is strong enough to distinguish Hilbert spaces from general Banach spaces. Thus Wasserstein barycenters provide an intrinsic optimal-transport test for Hilbertian geometry. More precisely, we show that if a finite-dimensional normed vector space, equipped with Lebesgue measure, satisfies the Wasserstein Jensen's inequality for the entropy at barycenters of arbitrary finite families of probability measures, then its norm must be induced by an inner product. This contrasts sharply with a well-known result: every finite-dimensional normed vector space satisfies the nonnegative Ricci curvature condition in the sense of Lott--Sturm--Villani, whereas barycenter convexity excludes all non-Hilbertian norms. As a consequence, smooth reversible Finsler manifolds satisfying the corresponding barycentric curvature-dimension condition have Riemannian tangent norms. The proof does not assume strict convexity of the norm. Its two main ingredients are a rank-one polarization argument, which yields the dual parallelogram identity in the strictly convex case, and a maximal-face trapping argument, which rules out flat faces of the unit ball.

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math.PR 2026-06-29

Wiener space transformations factor via time-ordered exponentials

by Jirô Akahori, Takafumi Amaba +1 more

Factorization of Time-Ordered Exponentials for Wiener Space Transformations

The explicit factorization into determinant, multiplication operator, and translation recovers classical formulas like Ramer-Kusuoka upon ta

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We develop an operator-algebraic framework for change-of-variables formulas on Wiener space, interpreting them as arising from hidden symmetries acting on observables. We show that general transformations can be represented by time-ordered exponentials generated by annihilation and creation operators, and that these admit an explicit factorization into a determinant, a multiplication operator, and a translation operator. Taking expectations recovers the classical formulas, including the Ramer--Kusuoka formula.

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math.FA 2026-06-29

Square cone factors maps through Lorentz sums with symmetric targets

by Guillaume Aubrun, Francesca La Piana +1 more

Factorization through Lorentz cones

The Lorentz factorization property holds exactly for the square-based cone paired with any symmetric cone, and fails for equal or polyhedral

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A pair of proper cones $(\mathsf{C}_1,\mathsf{C}_2)$ is said to have the Lorentz factorization property (LFP) if every $(\mathsf{C}_1,\mathsf{C}_2)$-positive map factors through a direct sum of Lorentzian cones, i.e., cones over Euclidean balls. Clearly, $(\mathsf{C}_1,\mathsf{C}_2)$ has the LFP if either $\mathsf{C}_1$ or $\mathsf{C}_2$ is a direct sum of Lorentzian cones, and our main goal is to find other examples. We show that such examples cannot be found for pairs $(\mathsf{C}_1,\mathsf{C}_2)$ where $\mathsf{C}_1=\mathsf{C}_2$, or in the case where both $\mathsf{C}_1$ and $\mathsf{C}_2$ are polyhedral. We also focus on the case where $\mathsf{C}_1=\mathsf{C}_\square$ is the square-based cone in $\mathbf{R}^3$. Here, we show that $(\mathsf{C}_\square,\mathsf{C})$ has the LFP whenever $\mathsf{C}$ is a symmetric cone, i.e., a direct sum of Lorentz cones, cones of positive semidefinite matrices over the real numbers, complex numbers or quaternions, and the cone of $3\times 3$ positive semidefinite matrices over the octonions. We leave open the question whether there are more examples, but we show that this list cannot be extended by any strictly convex cone $\mathsf{C}$ or for a cone $\mathsf{C}$ with $\text{dim}(\mathsf{C})\leq 5$. Finally, we discuss an application to a problem in quantum information theory.

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math.FA 2026-06-29

L^p triangle inequality sharpened by optimal stability term

by Ruizhou Song

Stability Refinements of the Triangle Inequality in L^p Spaces

Refinement subtracts a term based on normalized function difference, with optimal constants for positive functions when p exceeds 2.

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Let $(X,\mu)$ be a measure space and let $1< p< \infty$. We study quantitative stability refinements of Minkowski's inequality \[ \| f + g\|_{p}\leq \| f\|_{p} + \| g\|_{p} \] for real-valued functions in $L^p(X,\mu)$. We first establish a stability estimate for arbitrary real-valued functions and show that its constant is sharp. We then prove that, for nonnegative functions, the constant can be improved when $p\geq 2$, again to its optimal value. More precisely, if $f,g\geq 0$ and $f,g\neq 0$, then \[ \| f + g\|_{p}\leq \| f\|_{p} + \| g\|_{p} - c_{p}\min \{\| f\|_{p},\| g\|_{p}\} \left\| \frac{f}{\| f\|_{p}} -\frac{g}{\| g\|_{p}}\right\|_{p}^{\alpha_{p}}, \] where \[ c_p = \begin{cases} \dfrac{p-1}{4}, & 1<p\leq 2,\\[6pt] \dfrac{1-2^{1-p}}{p}, & 2\leq p<\infty, \end{cases} \qquad \alpha_p = \begin{cases} 2, & 1<p\leq 2,\\ p, & 2\leq p<\infty. \end{cases} \] Both constants are best possible.

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math.FA 2026-06-26

Review links continuous diffraction to vanishing Fourier-Bohr coefficients

by Nicolae Strungaru

Continuous diffraction spectrum and the uniform vanishing of Fourier--Bohr coefficients

The connection runs through consistent phase frequency and clarifies when spectra lack discrete peaks.

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In this paper we review the connection among continuity of the diffraction spectrum, the (uniform) vanishing of the Fourier--Bohr coefficients and the so called consistent phase frequency.

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math.AG 2026-06-26

Positivity certificates hold uniformly for symmetric functions in any dimension

by Sebastian Debus, Robin Schabert

Any-dimensional Positivstellens\"atze for symmetric functions

Two theorems extend the Pólya and Reznick results by tying truncated power sums to moments on [-1,1] and describing the infinite orbit space

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Positivstellens\"atze provide certificates of positivity for polynomials. Extending these certificates to symmetric functions, uniformly across all dimensions, presents structural challenges. For instance, the underlying domain is not semialgebraic. In this paper, we prove two Positivstellens\"atze for symmetric functions that are uniformly bounded below by some $\varepsilon > 0$. These are infinite dimensional analogous of theorems of P\'olya and Reznick. The proof relates evaluations of the (truncated) power sum map $(p_2,p_3,\dots)$ to moments of discrete probability measures on the compact interval $[-1,1]$. This yields a characterization of the orbit space of the infinite symmetric group. Finally, we provide an alternative proof of existing Positivstellens\"atze for normalized symmetric functions.

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math.FA 2026-06-26

Interpolation spaces get 2-rotund norms with uncountable bases

by Stephen DIlworth, Denka Kutzarova

2-rotundity of some nonseparable abstract interpolation spaces

Generalized from reflexive spaces and Schreier families on uncountable sets under mild complexity assumptions on the families.

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Generalizing a construction due to Argyros and Motakis \cite{AM}, we define a nonseparable abstract interpolation space associated to any given reflexive space with an unconditional basis together with the Schreier spaces associated to an increasing sequence of compact families of finite subsets of an uncountable set. Under a mild complexity assumption on the families, we prove that the interpolation space admits a $2$-rotund norm with an uncountable $1$-unconditional basis.

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math.NA 2026-06-26

Quasi-Feynman formulas give quadratic Chernoff approximations to parabolic PDEs

by Ivan D. Remizov, Alexandr V. Vedenin

Quasi-Feynman formulas that provide fast converging Chernoff approximations to solution of parabolic differential equation on the real line

New integral-based operator yields uniform quadratic-rate convergence on the real line, improving the usual first-order bound.

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We construct explicit approximations to the solution of a second-order parabolic partial differential equation on the real line with variable coefficients. The method is based on Chernoff's product formula and uses a new operator-valued function defined through proper Riemann integrals over a bounded interval, which makes the approach readily usable in numerical practice. For sufficiently smooth initial data and coefficients, we prove that the resulting Chernoff approximations converge uniformly in space and time with a quadratic rate, improving the standard first-order estimate. The construction yields a new class of quasi-Feynman formulas that are neither grid-based nor Galerkin-type, but instead rely on semigroup theory and multiple bounded integrals. The theoretical findings are validated by symbolic computation, and the paper contributes both refined error bounds and a practical analytical tool for parabolic problems.

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math.FA 2026-06-26

Property A implies p-nuclearity of ℓ^p uniform Roe algebras

by Yeong Chyuan Chung

On some p-approximation properties of exact discrete groups and ell^p uniform Roe algebras

For discrete spaces with bounded geometry this links coarse geometry to algebraic approximation for every p in (1, ∞).

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We study $p$-approximation properties of $\ell^p$ uniform Roe algebras and their connections to coarse geometry and group theory. For a discrete metric space $X$ with bounded geometry, we prove that property A implies $p$-nuclearity of the $\ell^p$ uniform Roe algebra $B^p_u(X)$ for every $p\in(1,\infty)$, while $B^1_u(X)$ is always 1-nuclear. We introduce the $p$-invariant translation approximation property ($p$-ITAP) for discrete groups, generalizing the 2-ITAP of Roe. We also introduce the $p$-operator ITAP. For exact groups, we show that the $p$-operator ITAP is equivalent to the $p$-approximation property of An-Lee-Ruan. We also characterize exactness of discrete groups in terms of their $\ell^p$ uniform Roe algebras with coefficients in $p$-operator spaces.

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math.FA 2026-06-26

Dobrakov integral yields generalized derivatives for vector measures

by Artem Yurievich Dudko

Towards a Theory of Dobrakov-Sobolev Spaces

Leibniz and integration-by-parts rules are proved for Banach functions against Fomin-differentiable operator measures, allowing Sobolev-type

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The aim of this paper is to introduce a generalization of Sobolev spaces based on the Dobrakov integral. More precisely, we consider the setting of Banach-valued functions and Fomin differentiable Borel operator-valued measures on a finite-dimensional space. To build the necessary rigorous foundation, we establish analogs of several key results from the theory of differentiable real-valued measures, including the Leibniz rule and the integration by parts formula, all within the context of Dobrakov integration. These results are then embedded into the general scheme of vector-valued distribution theory. In particular, we describe the configuration of test spaces that yields an appropriate definition of a generalized derivative with respect to a differentiable operator-valued measure.

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math.NA 2026-06-26

VP method gives stable solution for singular Fredholm equations

by Domenico Mezzanotte, Donatella Occorsio +2 more

De la Vall\'ee Poussin type approximation for solving some Fredholm integral equations

De la Vallée Poussin approximations at Jacobi zeros converge with higher local accuracy than Lagrange projections for kernels with singulari

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In the present paper, we introduce a numerical method for second-kind Fredholm integral equations (FIEs) based on de la Vall\'ee Poussin-type (VP) polynomial approximations at Jacobi zeros. This class of approximations offers several advantages over classical Lagrange interpolation at the same nodes. In particular, it guarantees uniformly bounded Lebesgue constants in suitable weighted function spaces and provides near-best uniform approximation for functions in these spaces, while also significantly mitigating the Gibbs phenomenon. We show how these properties can be exploited in the numerical solution of FIEs. In particular, the proposed approach effectively handles functions with possible algebraic endpoint singularities and kernel functions featuring weak singularities or highly oscillatory behavior. Under suitable assumptions, we prove stability and convergence of the method in weighted uniform spaces. Furthermore, we develop an efficient implementation based on the solution of a well-conditioned linear system. Numerical results confirm the theoretical error estimates and show that the proposed method achieves higher local accuracy than the corresponding Lagrange-based projection method.

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math.AP 2026-06-26

Deficit controls H^{1/2} boundary deviation for torsion maximizers

by Luca Barbato, Francesco Salerno

Local stability for a class of Saint-Venant type inequalities

Among domains close to a ball the gap to the optimal value bounds the square of the perturbation size measured in the H^{1/2} norm.

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We establish a local stability result for a class of Saint-Venant type inequalities. Given the solution $u$ of the Dirichlet torsion problem in a domain $\Omega$, we consider shape functionals $\mathcal{J}(\Omega)$ involving the integral of $j(u)$, where $j$ is convex and satisfies suitable structural assumptions. By Talenti's comparison principle, balls maximize $\mathcal{J}$ among sets of prescribed measure. We prove that this extremal property is stable in the class of nearly spherical sets: the deficit from the optimal value controls the square of the $H^{1/2}$-norm of the boundary perturbation. The argument relies on shape derivative techniques, including the computation of the second variation and the introduction of an adjoint state. As applications, the result covers several relevant examples, including the torsional rigidity, $L^p$-norms of the torsion function for $p\ge 2$, and Moser-Trudinger functional in dimension two.

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math.FA 2026-06-26

Plancherel identity holds for lattice-invariant unbounded subsets of R^d

by Dorin Ervin Dutkay, Piyali Chakraborty

Plancherel Identities for unbounded subsets of mathbb R^d

Pairs translated by dual full-rank lattices admit an isometric isomorphism under the restricted Fourier transform.

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We present a class of pairs of subsets of $\mathbb R^d$ for which the Fourier transform, when restricted to these subsets, is an isometric isomorphism, and thus the Plancherel identity is satisfied. The sets are invariant under translations by dual full-rank lattices.

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math.FA 2026-06-26

Duality preserves Morse critical groups for ratio convex functions

by Dong Zhang

Hidden critical and Morse equivalence behind duality: Theory and Applications

Polarity dual keeps sublevel homotopy, critical groups, and handle decompositions unchanged for RC functions and yields a decomposition-free

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The aim of this paper is to establish critical duality theory for ratios of nonnegative homogeneous convex functions (shorten for RC functions) and differences of convex functions (abbreviated as DC functions) on Banach spaces. Specifically, we establish a series of duality results on critical point theory and Morse theory for RC functions, including the homotopy type of sublevel sets, the Morse critical points and their Rothe critical groups, Lagrange critical points and their multiplicities, Lusternik-Schnirelman min-max critical values, Poincare polynomials, as well as the structure of handlebody decompositions, all of which are proved to be preserved under polarity dual. Moreover, we obtain the first critical duality theory of DC functions which does not depend on the DC decomposition. This answers a question left open from the work of Toland on DC functions and the work of Le-Pham on DC programming. We apply these results to provide a reformulation of the graph Cheeger constant using zonotopes; we introduce the contact data which serves as a geometric characterization of Lagrange criticality; and we show that the eigenproblems for 1-Laplacian and $\infty$-Laplacian on hypergraphs are equivalent to the contact problems of zonotopes, which indeed establishes a new characterization of zonotopes. We also prove a duality equivalence for certain nonlinear eigenvalue problems and bifurcation problems. Our study here reveals an intricate interaction of critical point theory with other fields such as convex analysis, combinatorial geometry, and nonlinear eigenproblems on graphs.

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math.NA 2026-06-26

Regularizations of metric generalized inverse diverge outside domain

by Peter Mathé, Bernd Hofmann

Regularization of the metric generalized inverse in Banach spaces and the dichotomy phenomenon

Approximations converge to best solutions inside the domain but grow unbounded for inputs outside it in strictly convex reflexive Banach spa

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For a bounded linear operator acting between Banach spaces, its metric generalized inverse is the analog to the prominent Moore-Penrose inverse for operators acting between Hilbert spaces. This generalized inverse is well-defined for Banach spaces that are strictly convex and reflexive. Previous studies had been restricted to closed range situations, where the metric generalized inverse constitutes a continuous homogeneous mapping. The focus of the present study is on the ill-posed situation in the sense of Nashed, when the governing operator has a non-closed range. We define and analyze iterative schemes as the Landweber and the Schulz-Newton method, as well as parametric schemes with focus on a specific Tikhonov method. Both types are called regularizations aimed at approximating the metric generalized inverse. As a fundamental feature of such schemes we observe a dichotomy. This emphasizes that these schemes, when applied to elements of the domain of the metric generalized inverse, approximate the corresponding best approximate solutions well, whereas the resulting approximations will be asymptotically unbounded if they are applied to elements that do not belong to the domain of the metric generalized inverse.

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