Papers

We prove stability estimates for the volume of central hyperplane projections of unit balls in $\ell_p^n(\mathbb{R})$, establishing the dual counterpart of the result of Chasapis, Nayar and Tkocz on stability of sections.

In this paper, we introduce a new class of operators, called $q$-Berezin sectorial operators, as an extension of the class of $q$-sectorial operators. By presenting examples on the Hardy-Hilbert space, we show that there exist operators which are $q$-Berezin sectorial but not $q$-sectorial. Several new inequalities for the $q$-Berezin number associated with this class of operators are also derived. In addition, we investigate the geometric structure of the $q$-Berezin range for various classes of operators on the Bergman space, including weighted shift and certain composition operators.

For every fractional order strictly between zero and one and every finite interval, there is a bounded representing function whose Riemann–Liouville fractional integral satisfies the weighted endpoint condition proposed by Vainikko but fails the absolute Marchaud condition almost everywhere. One bounded example therefore disproves the necessity implication in Vainikko's conjectured characterization simultaneously for every exponent strictly larger than one, including the essentially bounded endpoint. A different sine-series construction gives, at each exponent strictly between one and infinity, failure at every interior point for one representative. For the bounded example, signed inversion remains valid for every finite exponent greater than or equal to one. The converse implication in Vainikko's conjecture is not decided here.

We extend the discrete gauge rigidity of time-scaled intertwining cocycles to operator networks over a connected manifold M and classify the holonomy that the continuous setting admits. For dissipative semigroups $S_x(t)=e^{-tA_x}$ linked by transport operators $K_\gamma S_{\gamma(0)}(t)=S_{\gamma(1)}(\lambda(\gamma)t)K_\gamma$ along paths $\gamma$, we prove that the scaling field carries no monodromy: $\lambda(x,y)=\tau(x)/\tau(y)$ for a continuous positive function $\tau$, unique up to a multiplicative constant, with multiplicativity forced by the intertwining relation rather than assumed. For regular cocycles the transport defines a bounded operator connection on a Hilbert bundle over M; the holonomy of a closed loop is confined to the commutant of the generator and decomposes into spectral sectors, on each of which it is adiabatic (Berry-Wilczek-Zee) transport twisted by an independent commutant-valued sector potential; for flat unitary cocycles the holonomy is classified by unitary representations of $\pi_1(M)$ in the commutant. An explicit rotating network over $S^1$ exhibits holonomy equal to the parity operator while the Berry connection one-form of every level vanishes identically: the monodromy is carried by $\mathbb{Z}_2$ orientation classes of Moebius eigenline bundles.

Given a real-valued function $f$, let $\mathcal{N}_f(\delta, Q)$ be the number of rational points with denominators at most $Q\ge 1$ in the $(\delta/Q)$-tubular neighbourhood of the graph of the function $f$. A heuristic predicts that the number of such points grows like the area of the neighbourhood provided that $\delta$ is big enough (in a suitable sense). Considerable efforts have been committed to prove this heuristic for regular curves. This culminated in the works by Vaughan & Velani~(2006) and by Huang~(2015) establishing an asymptotic expansion for $\mathcal{N}_f(\delta, Q)$ provided that $\delta\gg Q^{-1+\epsilon}$ for some $\epsilon>0$ when the map $f$ is, among other assumptions, twice continuously differentiable. The present work deals with the thus-far unexplored regime where minimal regularity conditions are imposed on the curve. More precisely, it is concerned with the case where the map $f$ is an a.s. realisation of the graph of Brownian motion. The main result establishes the existence of an almost sure asymptotic expansion for the counting function for all values of $\delta$, with the exception of a critical regime, thereby going well beyond the theory currently available for regular curves. A key ingredient in the proof is the derivation of the area heuristic, which relies on establishing the a.s. asymptotics of the area of the tubular neighbourhood of the graph of Brownian motion. This result has two main consequences: firstly, it completes the counting aspect of the theory of Diophantine approximation on the graph of Brownian motion initiated by Sprindžuk (1979). Secondly, it hints at the existence of a theory unifying the analysis of rational points near a curve on the one hand and, on the other, its local Hölder regularity and fine-scale oscillations. It thus builds a seemingly new bridge between Number Theory and Multifractal Analysis.

For a bounded linear operator $T$ on a complex separable Hilbert space $\mathcal{H}$ and a vector $x\in \mathcal{H}$, let $W_a(T,x)$ be the set of cluster points of the sequence $\{\langle|T^n|^{1/n}x,x\rangle\}_{n=1}^{\infty}$. We define the asymptotic numerical range and the asymptotic numerical radius of $T$, respectively, by $W_a(T):=\bigcup_{\|x\|=1}W_a(T,x)$ and $w_a(T):=\sup W_a(T)$. We prove that $W_a(T,x)$ is a compact interval for every $x\in\mathcal{H}$ and that $W_a(T)$ is a bounded interval, where intervals are allowed to be degenerate. Using the asymptotic numerical radius, we show that if there is a nonzero vector $x_0\in \mathcal{H}$ with $r(T,x_0)<w_a(T)$, where $r(T,x_0)$ denotes the local spectral radius of $T$ at $x_0$, then the subspaces $\overline{\{x\in\mathcal{H}:r(T,x)\le r(T,x_0)\}}$ and $\overline{\operatorname{span}\{T^n x_0:n\ge0\}}$, which are well known to be hyperinvariant and invariant for $T$, respectively, are both nontrivial. We also prove that $w_a(T)=r(T)$ for every hyponormal operator $T$, where $r(T)$ denotes the spectral radius of $T$. Finally, we investigate the connectedness of the set of WOT cluster points of the sequence $\{|T^n|^{1/n}\}_{n=1}^\infty$.

We study finitely generated maximal right ideals of the Banach algebra $\mathcal{B}(E)$ of bounded operators on a complex Banach space $E$. Every maximal right ideal is either fixed by a non-zero functional or contains the ideal of finite-rank operators; when $E$ is infinite-dimensional, each non-fixed maximal right ideal in fact contains the ideal of inessential operators. Using the elementary representation of finitely generated right ideals as lifting ideals $\operatorname{Lift}(T)=\{TU:U\in\mathcal{B}(E,E^n)\}$, where $T\in\mathcal{B}(E^n,E)$ for some $n\in\mathbb{N}$, we identify the exact operator-theoretic obstruction. The ideal $\operatorname{Lift}(T)$ contains the finite-rank operators precisely when $T$ is surjective, and it equals $\mathcal{B}(E)$ precisely when $T$ is right invertible. If $T$ is surjective but not right invertible, then $\operatorname{Lift}(T)$ is maximal exactly when the row operator $[T\ S]$ is right invertible for every $S\in\mathcal{B}(E)\setminus\operatorname{Lift}(T)$. We apply this framework, together with duality, pullback, lattice-theoretic and cardinality arguments, to obtain maximal right ideals which are not finitely generated for large classes of Banach spaces. These include the following infinite-dimensional spaces: reflexive spaces, separable spaces with an unconditional Schauder decomposition into a countably infinite sequence of non-zero subspaces, spaces containing a complemented copy of $\ell_1$, KB-spaces, Lebesgue spaces $L_p(\mu)$ for $1\leqslant p<\infty$, full Orlicz spaces with order-continuous norm, and scalar-plus-compact spaces. We obtain the stronger conclusion that every finitely generated maximal right ideal is fixed for Hilbert spaces, $\ell_1(\Gamma)$-spaces, reflexive spaces with the bounded approximation property, and the mixed spaces $\ell_1(\Gamma)\oplus H$ with $H$ a separable Hilbert space.

A Banach space $X$ has the SHAI property if, for every non-zero Banach space $Y$, every surjective algebra homomorphism from the algebra $\mathcal{L}(X)$ of bounded linear operators on $X$ onto $\mathcal{L}(Y)$ is injective. In this work, we prove that the Bourgain-Rosenthal-Schechtman spaces have the SHAI property, thereby answering a question posed by Johnson, Phillips, and Schechtman in 2022.

In a wide variety of fields, every non-scalar matrix $M$ is similar to a matrix all of whose entries are non-zero. We give extensions of this result, including some in the division ring context. We then apply these results to the question of which pairs of operators generate the algebra of all operators on finite- and infinite-dimensional spaces. In the case of complex, separable infinite-dimensional Hilbert space, this is intimately related to the unsolved Invariant Subspace Problem. Finally, linear independence of subsets of arbitrary TVS's and generators of arbitrary TA's are shown to be stable in the sense of the theorems presented.

We prove that any $n$-dimensional normed space is at Banach–Mazur distance $O(n^{2/3})$ from $\ell_\infty^n$.

Let $H$ be a real Hilbert space, let $K\subseteq H$ be a closed convex cone, let $K^*$ be its dual closed convex cone, and let $P_K$ denote the metric projection onto $K$. We study scalar quantities associated with th projection $P_K$ and their interaction with the preorder induced by $K$. Our main result characterizes subduality by the monotonicity of the norm of a projection: $K\subseteq K^*$ if and only if $x\le_K y$ implies $\|P_Kx\|\le \|P_Ky\|$. We also discuss the sublinearity of the norm of a projection and of the distance from a closed convex cone, relate these functions to asymmetric seminorms, and introduce a normalized angular distance from a closed convex cone. The angular distance admits equivalent formulas in terms of $P_K$ and the ordinary distance function and, for closed convex cones with $K\ne H$, yields a second characterization of subduality. Finally, coordinatewise monotone norms on finite-dimensional spaces are used to aggregate distances from several closed convex cones and thereby construct asymmetric seminorms. The proofs are consequences of Moreau's decomposition theorem and the geometry of metric projections.

We develop a systematic theory of uncertainty principles for the short-time Fourier transform (STFT) on the Heisenberg group. Building on recent developments in modulation spaces and time-frequency analysis on the Heisenberg group introduced by Fischer et al. and later by Biswas-Thangavelu, we establish noncommutative analogues of several fundamental uncertainty principles in time-frequency analysis, including Benedicks' theorem, the Donoho-Stark uncertainty principle, and Lieb's inequality. As a consequence of Lieb's inequality, we derive an entropy-based uncertainty principle of Hirschman type. We further establish a Heisenberg-type uncertainty inequality and local uncertainty principles in the spirit of Price. In addition, we prove a Beurling-Hardy-type theorem that captures the interplay between decay and phase-space localization. Finally, we investigate decay properties of the STFT and their implications for time-frequency concentration. These results extend a broad spectrum of classical uncertainty phenomena from the Euclidean setting to the Heisenberg group, highlighting the role of noncommutative harmonic analysis in the study of phase-space localization.

We prove a surprisingly simple pointwise formula for the Frobenius norm of the tensor of $n$-th order partial derivatives of a radially symmetric function $f(x)=g(r(x))$. Using the iterations of the differential operator $\mathcal{D} g(r)=g'(r)/r$, we avoid technical difficulties usually caused by higher-order radial derivatives. As a consequence, we obtain a complete characterization of the subspace of radially symmetric functions in both inhomogeneous and homogeneous Sobolev spaces of arbitrarily high order and all integrability parameters $p\in [1,\infty).$ Furthermore, the pointwise nature of our approach allows us to obtain similar results also for Sobolev-type spaces built upon more general Banach lattices.

For every $1<p\neq2<\infty$, we exhibit an explicit Schur multiplier on $S_p$ whose operator norm is strictly smaller than its completely bounded norm. More precisely, for each such $p$, we construct a finitely supported Schur symbol $m_p$ such that \[ \|M_{m_p}\|_{p\to p} < \|M_{m_p}\|_{\mathrm{cb},p}. \] This answers the question raised by Lafforgue and de la Salle in 2011 after their Conjecture~1.8 and gives an affirmative answer to Statement~2 in Section~5 of Caspers and Wildschut (2019). In particular, it disproves Statement~3 of Caspers and Wildschut (2019) throughout the same range of exponents.

We introduce the sup-inf-completion of a Dedekind complete vector lattice, an essentially unique extension in which every nonempty subset has both a supremum and an infimum. Since this completion is not a cone, we develop the more general framework of lattice stars, which provides the natural setting for its construction. We establish the fundamental properties of the sup-inf-completion, including a universal property, a representation theorem, and a characterization of its bands and band projections. As an application, we extend the Riemann integral on Dedekind complete $f$-algebras to Type I and Type II improper integrals. We conclude by showing that power series on universally complete vector lattices may be integrated term-by-term.

We show that uniform thin-shell estimates for isotropic log-concave measures $\mu$ on $\mathbb{R}^n$ yield precise and explicit deviation estimates for the Euclidean norm $|X|$ below the expectation, improving the square-root dependence of Klartag-Lehec to a quadratic one (which is best possible, up to numeric constants). Using the recent Chen-Klartag sharp variance bound, we deduce: $$ \mu\left(|X|\le \sqrt{n}-s\right) \le \exp\left\{ -\frac{s^2}{4} \left(1+O\left(\frac{s}{\sqrt{n}}\right)\right) \right\} \qquad \forall\, s\in(0,\sqrt{n}). $$ In particular, this yields a new and transparent proof that Thin-Shell implies Slicing (by passing through small-ball estimates). Our method is based on using central Gaussian tilts, recently introduced by Brazitikos, which may be thought of as a deterministic version of Eldan's stochastic localization. An anisotropic variant (when $\mu$ has general covariance structure) of these deviation estimates is also obtained.

We develop constructive approximation and learning guarantees for shallow neural models with infinite-dimensional inputs observed through finitely many coordinates. The analysis is based on a parameter-normalized neural dictionary and its associated weighted variation class. Within this class, the approximation error separates into a distribution-dependent coordinate-truncation term and a greedy finite-width term. For empirical regression, a fully-corrective greedy procedure yields population guarantees whose statistical complexity is uniform in the retained input resolution. The same framework extends to Hilbert-valued responses without an explicit dependence on the output dimension. The dimension-free statements are statistical, not computational: selecting a new neuron still requires solving a nonconvex parameter-search problem. The quasi-Polish construction underlying recent infinite-dimensional universal approximation results provides a motivating example, and synthetic experiments illustrate the predicted resolution, width, and sample-size regimes.

We establish a collection of sharp inequalities for real quadratic forms on the first-quadrant sector of a regular octagon. Starting from the complete extreme-point description of the associated polynomial unit ball, we compute the exact pointwise Bernstein function for the Euclidean gradient. One extreme curve controls the problem: its endpoint is active up to slope $1/2$, after which the maximizer follows an explicit Cardano branch. We obtain the sharp Markov constant $2\sqrt5$, the exact relative quadratic polarization constant $2$, the canonical unconditional constant $3$, and the body-relative Bohr radius $1/\sqrt3$. We also determine the optimal coefficient $\ell_q$-comparison for every $1\le q\le\infty$. The same norm-one polynomial is extremal for all these global constants. At $q=4/3$ the result is a sharp fixed-space coefficient inequality of Bohnenblust–Hille type.

Let $A\in\mathcal{B}(H)^+$ and $B\in\mathcal{B}(\ell^2)^+$ be positive bounded operators. We investigate reduced weighted adjoints of densely defined closable operators and use them to develop frame-type systems in the semi-Hilbert spaces induced by $A$ and $B$. Closedness, boundedness, range behavior, and algebraic properties of the $A$-adjoint are established, with particular attention to sums, products, and double adjoints in the unbounded setting. We then study the associated $AB$-analysis, $AB$-synthesis, and $AB$-frame operators. Operator-theoretic characterizations of $AB$-Bessel sequences, $AB$-lower semi-frames, and $AB$-frames are obtained, and examples show that several natural weighted-adjoint identities may hold only as proper inclusions. A Douglas-type range characterization for $AB$-frames is also derived.

We study frames and related sequences in tensor products of separable Hilbert spaces from an algebraic perspective. For sequences $\{f_n\}_{n\in\mathbb{N}} \subset H_1$ and $\{g_m\}_{m\in\mathbb{N}} \subset H_2$, we consider the tensor sequence $\{f_n \otimes g_m\}_{n,m\in\mathbb{N}}$ in $H_1 \otimes H_2$. We characterize the lower semi-frame and Riesz–Fischer properties of tensor sequences in terms of the corresponding properties of the component sequences. These results extend to infinite tensor products. In addition, we study the action of tensor sums of operators on frame-related sequences in tensor product Hilbert spaces.

We determine the exact norm of the Hilbert matrix operator $\mathcal H$ on the $\alpha$-Bloch space $\mathcal B^\alpha$ for $1<\alpha<2$, together with all norm-attaining functions of norm one. For $1<\alpha\leq3/2$, we obtain an explicit formula for the exact norm in terms of the Gamma function. For $3/2<\alpha<2$, we obtain an exact one-dimensional maximization formula and show that the corresponding maximum is attained at an interior point of $(0,1)$. We also determine the exact norm of $\mathcal H:\mathcal B^\alpha\to\mathcal B^\alpha_{\log}$. The norm formula changes at the critical parameter $\alpha=4/3$, and the exact norm is obtained for the full range $1<\alpha<2$. The norm-attaining functions are completely characterized for both operators.

We study the role of Banach space geometry in deterministic and stochastic maximal regularity. We first construct an example showing that the UMD assumption in the characterisation of maximal $L^p$-regularity in terms of $R$-sectoriality cannot be omitted. Combining this construction with an equivalence between stochastic maximal regularity and deterministic maximal regularity on the $2$-concavification of the underlying space, we obtain an operator on a UMD Banach function space of type $2$ that has a bounded $H^\infty$-calculus of angle zero, but fails stochastic maximal $L^p$-regularity (SMR$_p$) for every $p\in[2,\infty)$. Motivated by this example, we study the Banach space geometry hypothesis underlying SMR$_p$ more closely. This is an $R$-boundedness condition $(S_p)$ for stochastic convolution operators. For UMD spaces $X$ of type $2$, we show that this condition is not only sufficient, but also necessary for two canonical test operators: a diagonal multiplier on a Rademacher space and, for $q>2$, the Laplacian on $L^q(\mathbb R^d;X)$. Finally, we prove that its interval-kernel and exponential-kernel formulations are equivalent and that, at the endpoint $p=2$, condition $(S_2)$ holds if and only if $X$ is isomorphic to a Hilbert space.

Let $S_n$ denote the smallest constant such that \[ |\det T|\|T^{-1}\| \leq S_n \|T\|^{n-1} \] for every invertible operator $T$ on every $n$-dimensional complex Banach space. In Hilbert space the optimal constant is $1$. For arbitrary Banach spaces, J. J. Schäffer proved in 1970 that \[S_n\leq \sqrt{en}. \] Subsequent work showed that $S_n$ grows like $\sqrt n$, but the sharp asymptotic constant has remained open for more than five decades. We resolve this problem by proving \[ \lim_{n\to\infty}\frac{S_n}{\sqrt n}=\sqrt e. \] Thus Schäffer's upper bound is asymptotically sharp, including its constant. Our proof is constructive, providing explicit Banach-space norms through duality and explicit matrices through the theory of model operators. At the analytic core of the argument, an extremal formulation of Schäffer's problem in the Wiener algebra reduces the matching asymptotic lower bound for $S_n$ to uniformly controlling the Taylor coefficients of products $QB_n$, where $B_n$ is a finite Blaschke product of degree $n$ and $Q$ is a polynomial factor. We optimize simultaneously the zero distribution of $B_n$ and the choice of $Q$. The resulting zeros follow a logarithmic asymptotic distribution, and a sharp uniform asymptotic analysis of the Taylor coefficients of $QB_n$ yields the constant $\sqrt e$. The corresponding model operators then yield matrices with these spectra that asymptotically attain Schäffer's bound.

We investigate the relation between the complete isometry class of subspaces of noncommutative $\mathrm{L}^p$-spaces and their contractive complementability, where $1 < p < \infty$ with $p \not= 2$. We show that if $P \colon \mathrm{L}^p(\mathcal{M}) \to \mathrm{L}^p(\mathcal{M})$ is a positive contractive projection whose range is completely isometric to another noncommutative $\mathrm{L}^p$-space, then $P$ is necessarily completely positive. This provides a converse to the main result of [ArR24] and the positivity assumption on $P$ is essential. We further establish a rectangular analogue of this result. More precisely, we prove that every closed subspace of a noncommutative $\mathrm{L}^p$-space which is completely isometric to a rectangular noncommutative $\mathrm{L}^p$-space of the form $e\mathrm{L}^p(\mathcal{N})(1-e)$ is the range of a contractively decomposable projection. Combined with the known converse implication, this yields a characterization of the ranges of contractively decomposable projections as precisely the subspaces completely isometric to rectangular $\mathrm{L}^p$-spaces associated with $\mathrm{W}^*$-ternary rings of operators.

Let $ C \subseteq \mathbb{R}^n $ be a convex set. The mapping $ f : C \longrightarrow \mathbb{R}^n $ is strictly increasing, if $ \langle f(x)-f(y), x-y \rangle > 0 $ for all distinct elements $ x,y \in C $. Applying classical theorems of finite dimensional convex geometry and convex analysis, we show that the inverse functions has a unique extension $ f^{(-1)} : \mathrm{conv} (f(C)) \longrightarrow C $ such that $ f^{(-1)} $ is monotone, continuous and it acts as a left-inverse of $ f $. As an application we introduce the concept of vector-valued weighted quasi-arithmetic means and discuss their equality problem.

This paper is devoted to the study of the numerical range of linear relations in Banach spaces. We present a new definition adapted to the multivalued nature of linear relations and analyze its main properties. We establish spectral inclusion results showing that the spectrum is contained in the union of the closure of the numerical range of a linear relation and the numerical range of its Banach adjoint, together with resolvent estimates related to the distance to the numerical range. As an application, we derive spectral enclosures for operator pencils by associating them with suitable linear relations and introduce corresponding numerical ranges for operator pencils in Banach spaces. We show that this approach may provide sharper information than classical numerical ranges of operator pencils. Furthermore, we use the numerical abscissas to show that the Banach space numerical range can yield strict exponential decay for semigroups that cannot be obtained from the corresponding Hilbert space numerical range.

We consider the natural family of Fourier inequalities for the Paley–Wiener space $\mathrm{PW}^q(\Omega)$, consisting of $L^q$-functions with Fourier support in a convex set $\Omega \subset \mathbb{R}^n$, $n \geq 2$, free of affine lines. Namely, \[ \int_{\Omega}\dfrac{|\hat{f}(x)|^p}{\omega_{\Omega}^d(x)}dx\leq C\|f\|_{L^q}^p,\quad f\in \mathrm{PW}^q(\Omega). \] Here $\hat{f}$ is the Fourier transform of $f$, $1 \leq p, q < \infty$, $d \in \mathbb{R}$, and $\omega_\Omega$ is the frequency multiplier weight associated with the Paley–Wiener space of $\Omega$, \[ \omega_{\Omega}(x)=m(\Omega\cap (2x-\Omega)), \qquad x \in \Omega. \] For an arbitrary polyhedron $P$, we completely characterize the triples $(p,q,d)$ which yield valid Fourier inequalities. For a ball $B$, we characterize the valid triples when $p \geq 2$. When $p < 2$, the situation is different for the ball, and natural critical inequalities fail. However, we show that the spherical restriction conjecture implies a family of subcritical Fourier inequalities for the ball, which in turn imply the Kakeya conjecture (in its Minkowski-form). Finally, we link our family of Fourier inequalities to the theory of truncated Hankel operators acting on the Paley–Wiener space of $\Omega$.

We establish upper and lower bounds for relative entropy contraction of generalized quantum depolarizing channels and semigroups. Our bound provides tight first order asymptotic of the contraction rate in terms of the dimension constant. One side estimate are based on sharp reverse ratio and convexity of relative entropy of two states, which can be derived from the recently introduced Hockey-Stick quantum $f$-divergence, and also independently, Bogoliubov–Kubo–Mori quantum Fisher information metric. The other side follows from the existence of index achieving pure state with respect to a general conditional expectation. Our results extend to the complete entropy contraction rate, tensor stable estimates for product dynamics. Examples include quantum depolarization, dephasing, and compact group symmetrization, with consequences for the decay rate of coherence and asymmetry.

We investigate the existence problem of frame measures for uniform consecutive-digit Bernoulli convolutions under the separated scaling condition. Given parameters $N\geq 2$ and $0<\rho<1/N$, we prove that if $\rho^{-m}=B$ for some integers $m\geq 1$ and $B\geq 2$, with $N\nmid B$, then the associated $N$-Bernoulli convolution $\mu_{\rho,N}$ admits no frame measure. The proof introduces a new cyclic mask-quotient obstruction adapted to the multi-character structure of the consecutive-digit framework.

We prove the cyclic-product conjecture of Holmes, Holroyd and Ramírez: among all circular arrangements of a decreasing positive sequence, the circular-symmetric “greedy” arrangement maximises every consecutive cyclic product sum $P_r$.