Papers

We study compactness of the product of two weighted Hardy primitives into $L^q(u)$, $0<q<\infty$, on the input boundary $1\le p_1,p_2\le\infty$, where at least one input exponent is $1$ or $\infty$. For $L^1\times L^p$, $1<p<\infty$, boundedness implies compactness when $q<p/(p+1)$. For $L^1\times L^1$, the corresponding range is $q<1/2$. At and above these boundaries compactness is characterized by additional active-tail decay, and critical power weights give bounded noncompact examples. The endpoint reductions use cumulative Stieltjes measures with their initial atoms and strict Hardy-Copson brackets. An output-mass allocation argument proves local finite-rank density, including quasi-Banach targets. Applying the localization principles developed for the interior-input regime identifies the essential distance, finite-rank distance and Hausdorff noncompactness radius with the coupled active tail, with the appropriate metric power when $q<1$. For a finite left endpoint, ordinary $L^\infty$ inputs admit an exact scalar transfer completed by linear nodal approximation.

This paper is concerned with the effect of spatial heterogeneity on the linear stability of unique positive steady states for a nonlocal population equation. The main difficulty is that, because of the nonlocal term, the linearized operator need not be self-adjoint or satisfy the maximum principle, so the usual stability arguments based on the principal eigenvalue do not apply directly. We establish linear stability for sufficiently large diffusion and, under suitable conditions on the maximum set of the local growth ratio, for sufficiently small diffusion. In one space dimension, we also construct unstable positive steady states and obtain Hopf bifurcations for sufficiently small diffusion. Keeping the local coefficients fixed, we obtain either a supercritical or a subcritical Hopf bifurcation by changing the weight in the nonlocal integral. Our results show that uniqueness of a positive steady state does not imply its linear stability.

Let a normalized continuous Parseval frame describe the phase space of a reproducing kernel Hilbert space. The frame coefficients of an orthonormal family have total mass equal to the number of states and pointwise mass at most one. This elementary exclusion principle gives explicit lower bounds for the total cost of a family in terms of the volume of low-cost regions. For radial costs, it yields collective mean-dispersion, umbrella, and transport inequalities. The same argument applies to normalized Bessel families, with the Bessel bound measuring the permitted overlap. We also consider semiclassical families for which the frame measures vary. Weak convergence of the normalized measures, together with averaged localization of the normalized kernels, yields continuous-symbol Szegő limits and sharp asymptotics for the lowest Toeplitz energies. The results apply to Bargmann–Fock and weighted Bergman spaces, Gabor and Paley–Wiener spaces, and orthogonal-polynomial and determinantal ensembles, including the GUE.

Unital quantum channels are the noncommutative analogues of doubly stochastic matrices. In 1959, Marcus and Ree showed that the squared Frobenius norm of a doubly stochastic matrix is at most its maximal diagonal sum. We prove a quantum version of this inequality for mixed unitary channels on $\mathbb{M}_d$ and give an example showing that it fails for general unital channels. As in the classical case, we call the channels attaining equality as Erdős channels. We show that the trace-preserving conditional expectation onto any unital $C^*$-subalgebra of $\mathbb{M}_d$ is an Erdős channel. It is also locally isolated among channels of the same Choi rank. Next, we introduce quantum RCDS channels. These are the noncommutative analogue of doubly stochastic matrices whose supported permutations all have the same diagonal sum. We prove a structure theorem for them and use it to characterize the quantum RCDS Erdős channels. We give an explicit form, up to unitary equivalence, for the Erdős channels of Choi rank $2$, and further show that there are uncountably many unitary equivalence classes of Erdős channels unlike the classical case. We also characterize the Erdős channels among mixed Weyl-unitary channels of any Choi rank. For qubits, we prove that there are exactly four unitary equivalence classes of Erdős channels and they are represented by the identity channel and uniform averages of Pauli channels. Finally, restricting to the diagonal channels, our results recover the Marcus–Ree inequality, the structure of RCDS doubly stochastic matrices, and the characterization of Erdős.

We study combinatorial Banach spaces, i.e. Banach spaces induced by families of finite sets, of uncountable density. We prove some general theorems, for example we characterize when such spaces contain isomorphic copies of long $c_0$ and long $\ell_1$. We discuss the existence of Banach spaces which are (complementedly) universal in the category of Banach spaces with unconditional basis of fixed cardinality. We present some examples, in particular we show that the Banach spaces generated by Suslin trees do not contain uncountable equilateral sets.

Let $K\subset\mathbb{R}^n$ be a convex body, let $X$ and $Y$ be independent uniform points of $K$, and set $R_K=(|K|/\omega_n)^{1/n}$. We consider the distribution function \[ D_K(\rho)=\mathbb P\{|X-Y|/R_K<\rho\}, \] and its homogeneous counterpart \begin{equation*} \mathcal J_\rho(K)=\iint_{K\times K}\mathbf{1}_{\{|x-y|<\rho R_K\}}dxdy. \end{equation*} Thus $D_K(\rho)$ is the proportion of ordered pairs of points of $K$ whose distance is less than $\rho R_K$, or equivalently the radial distribution function of the normalized covariogram of $K$. The functional satisfies a Brunn–Minkowski inequality, with equality precisely for homothetic bodies, and its differential gives a first Minkowski inequality. We characterize the Borel measures that occur as Wulff differentials: they are exactly the nonzero finite positive measures with zero centroid which are not concentrated on a great subsphere, and the corresponding body is unique up to translation. We further prove fixed-volume and stationary rigidity of the ball in the subcritical range, together with regularity for the resulting equation. At saturation, the equation is the classical Minkowski equation.

Our study connects root-mean-square ballistic transport to potential-theoretic recurrence for finite-range open quantum random walks. In homogeneous walks with primitive local channels, finitely many simple periodic Fourier peripheral eigenvalues, no nonzero stationary Fourier mode, and a nondegenerate quadratic spectral term, we prove a periodic uniform local limit theorem and recover exponential finite-set return bounds for nonzero drift. The centered case yields strong Green-function asymptotics in at least three dimensions, as well as potential-kernel asymptotics in one and two dimensions. According to these assumptions, the RMS ballistic speed equals the drift norm, nonzero drift indicates transience, and centered walks are recurring in effective dimensions one and two but transitory in higher dimensions. The low-dimensional finding shows a recurrence of the origin projection in TOM. For reducible walks, an explicit harmonic \(h\)-transform converts each absorption component to an OQRW, providing a detailed breakdown of Green occupation potentials. The squared RMS speed is the absorption-weighted mean of squared component drifts, but the reduced drift-dimension classification also needs specific component return estimations. A centered noncommuting family validates the fundamental spectral assumptions in all dimensions and provides explicit potential constants. Exact finite traps and sparse reflecting barriers provide a complementary nonhomogeneous method for zero speed and TOM recurrence.

We prove two deterministic results for families of distances arising in the study of canonical processes. The first derives an admissible partition scheme from a growth condition. The second gives a representation in terms of parameterized separation trees and compares it with the corresponding majorizing-measure quantities. The main point is that the proofs do not depend on the distribution of the underlying process: once the initial distance and the family of distances are given, no random variables, independence, tail functions, or moment estimates are used. For canonical processes with regular log-concave tails, the assumptions of the abstract results follow from the usual regularity conditions. One direction of the separation-tree estimate also applies to Bernoulli processes without these additional assumptions, and we prove the reverse estimate for bounded convex unconditional index sets. We also give a version of the growth argument for points which, for finite index sets, leads to a recursive construction of admissible partitions and parameterized separation trees.

We classify the set of finite simple undirected graphs whose corresponding graph operator system is C*-nuclear. We prove that the operator system associated to the graph is C*-nuclear if and only if the graph is 2-chordal, as defined by McCullough. In particular, we present the first known example of a non-chordal graph whose graph operator system is C*-nuclear, as well as the first known examples of graph operator systems which are not C*-nuclear.

We study the Banach space and operator factorization structure of Janson-Sobolev and Janson-Sobolev-Hardy spaces. This new class of martingale spaces is determined by a $q$-adic filtration, a subspace $V\subset \mathbb{R}_0^{l\times q}$, and a rearrangement invariant function space $X$. Our main result shows that, for every bounded diagonal operator $D$, the operator $S = \sum_{t=1}^s \lambda_{\mathcal U}^{k_t}(D)Q_{\mathcal K_t}^{\mathcal B}$ determined by the linear functionals $\lambda_{\mathcal U}^{k_t}(D)$ and the canonical projections $Q_{\mathcal K_t}^{\mathcal B}$, almost projectionally factors through $D$ with constant $1^+$. As consequences, we obtain factorization results for diagonal operators both under a natural boundedness condition on the canonical projections and for all spaces equipped with the $L^1$-norm.

We study finite-time blow-up and global existence for semilinear heat equations with time-dependent reaction coefficients on metric measure spaces. For convex nonlinearities, we establish a blow-up criterion that retains the dependence on the initial datum through its linear evolution and does not require stochastic completeness. Under additional conservativity and heat kernel assumptions, we also prove blow-up at the critical Fujita exponent associated with polynomial volume growth. Our main global existence result is based on a heat-kernel-weighted functional space in which the mild solution map becomes a contraction. This yields a unified construction covering both polynomial and exponential heat kernel decay, general locally Lipschitz nonlinearities, and time-dependent reaction coefficients, without convexity or monotonicity assumptions. In particular, the global existence criterion only uses the behaviour of the nonlinearity on the range explored by the solution. Applications include Riemannian manifolds, metric graphs, and, within a separate discrete framework, weighted graphs. The arguments, especially the functional construction underlying global existence, are new and lead to new results even for the classical underlying spaces considered above.

We study sparse Kahane–Salem–Zygmund constructions and weighted Hardy–Littlewood inequalities for homogeneous polynomials. For supports of cardinality $n^{d+o(1)}$, we determine the sharp power of $n$ governing the smallest norm of a unimodular $m$-linear form on $\ell_{p_1}^n\times\cdots\times\ell_{p_m}^n$; in the diagonal case, this yields the missing polynomial growth exponent in the coefficient-versus-supremum norm problem for $2\le p\le m$ and $2\le r\le\infty$. We then introduce a diagonal weighted Hardy–Littlewood functional which, on $s=p\ge m$, agrees exactly with the classical Hardy–Littlewood coefficient norm with the same optimal constant. We determine the optimal diagonal weight exponent for $2\le p\le m$ and $1\le q\le2$, on the full critical line $p=m$, and on a sharp part of the region $q>2$; at $q=\infty$ the optimal weight exponent is obtained for every $2\le p\le m$. The sparse coefficient estimates provide the matching dimensional obstructions.

We prove stability of concentration in the Paley-Wiener space for each fixed time-bandwidth product. The concentration deficit controls the squared $L^2(\mathbb R)$-distance from a translated first prolate spheroidal wave function and the squared normalized symmetric difference from an interval at the same center. The obtained exponent for the function distance is optimal. The main idea is the introduction of the set discrepancy weighted by the best optimizers in place of the symmetric difference. The proof then combines elementary Hilbert space arguments with classical properties of the prolate spheroidal wave functions.

We study the topological dynamical notion of asymptotic expansivity for linear operators. We also introduce two new notions, namely strong asymptotic expansivity and super asymptotic expansivity. We show that all three notions are equivalent to hyperbolicity for linear operators on finite dimensional spaces. In case of infinite-dimension, we construct examples to show that asymptotic expansivity is weaker than strong asymptotic expansivity which is further weaker than super asymptotic expansivity. More interestingly, we prove that super asymptotic expansivity is equivalent to the notion of hyperbolicity. Then, we prove that the super asymptotically expansive linear operators is a dense class in strong operator topology. Finally, we show elegant ways to construct variants of asymptotically expansive linear operators.

We characterize extreme points of the (positive) unit balls of Marcinkiewicz spaces with respect to their natural quasi-norms. Having these results at hand, we show that every positive surjective isometry on a noncommutative Marcinkiewicz space is of elementary form. In particular, a linear mapping on a noncommutative Marcinkiewicz space over an atomic von Neumann algebra with all atoms having the same trace is a positive surjective isometry if and only if it is the restriction of a Jordan $*$-isomorphism which preserves the singular value functions. We also study (not necessarily positive) surjective isometries on weak $L_p$-spaces affiliated a $\sigma$-finite non-atomic von Neumann algebra for $1<p<\infty$, Marcinkiewicz sequence spaces, and weak $\ell_p$ operator ideals, $p>0$, as well as one-parameter groups of isometries on the latter two.

We show that the spectral radius introduced by Shalit and Shamovich uniquely determines the underlying operator space structure: If E_1 and E_2 are operator space structures on C^d such that the corresponding spectral radii \rho_{E_1} and \rho_{E_2} agree on every d-tuple of operators, then the identity map is completely isometric. This answers a question of Scherer, Shalit and Shamovich. Moreover, we obtain an analogous result for the minimal spectral radii, and showcase an application to the completely bounded isomorphism problem for algebras of bounded non-commutative functions on operator space unit balls.

Let $L=-\Delta_k+V$ be a Schrödinger operator associated with the Dunkl Laplacian and a nonnegative potential. We characterize the Hardy space $H_L^1$, defined by the maximal operator of the semigroup generated by $-L$, in terms of the Riesz transforms associated with $L$. Under a local integral condition on the potential and a decay condition on the semigroup, formulated relative to an admissible covering by cubes, we prove that \[ \|f\|_{H_L^1} \asymp \|f\|_{L^1(dw)} +\sum_{j=1}^N \|\widetilde{R}_j f\|_{L^1(dw)}. \] The proof combines an atomic characterization of $H_L^1$ with local Hardy space theory and weighted integral estimates for Dunkl derivatives of the heat kernels. The result applies to arbitrary root systems and nonnegative multiplicity functions.

In this paper, in addition to establishing several new sufficient or equivalent conditions for the well-known Separable Quotient Problem (SQP), we show that every infinite-dimensional Banach space lacking the Lotz property admits a separable quotient, thereby revealing a surprising connection between the SQP and semigroup theory.

We generalise Berger's dilation theorem from numerical ranges to scaled $q$-numerical ranges. Specifically, we prove that if the scaled $q$-numerical range $\frac{W_q(A)}{q}$ is contained in the closed unit disk, then $A$ admits a unitary $\rho_q$-dilation, where $$ \rho_q=\max\left\{1,\frac{2|q|}{1+\sqrt{1-|q|^2}}\right\}. $$ As a corollary we show that every disk containing $\Omega_q(A)$ is a complete $\rho_q$-spectral set for $A$. At $q=1$, this recovers the classical fact that any disk containing $W(A)$ is a complete $2$-spectral set for $A$.

We show that input-to-state stability (ISS) and integral input-to-state stability (integral ISS) with respect to the space of essentially bounded functions are equivalent properties for linear systems. The proof combines recent results on Orlicz-admissibility and a duality result due to the authors. As a direct consequence, we show that, given an $L^\infty$-admissible control operator, the mild solutions of the linear system are continuous, resolving a question by G.\ Weiss.

We describe hearts of strongly deflation-exact categories with kernels as localizations of categories of three-term complexes up to homotopy, without assuming admissibility of kernels. We construct a calculus of fractions and give explicit formulas for kernels and cokernels. For complete locally convex spaces, whose failure to be quasiabelian was established by Prosmans (2000), our description and its dual provide derived equivalent abelian models through two choices of a notion of exactness. For quasi-complete, sequentially complete and locally (a.k.a. Mackey) complete spaces, we show that the topologically exact sequences form the maximal deflation-exact structure, although these categories are not extension-closed in the category of locally convex Hausdorff spaces. Our results make them nevertheless accessible to homological algebra with respect to the induced notion of exactness. Further applications concern the quasiabelian category of bornological modules equipped with the non-maximal linear split exact structure.

Open quantum systems can rapidly lose most microscopic information, leaving only a few degrees of freedom to govern their long-time dynamics. Classical Markov state models (MSMs) describe such metastable dynamics as transitions among a few representative phases and are widely used to reduce complex-system dynamics in condensed matter and chemical physics. In quantum systems, however, phase labels alone are insufficient when coherence persists between metastable states. Even when the slow modes are known, their spectral projection need not produce valid quantum states. We construct quantum Markov state models (QMSMs) that describe the surviving information and its evolution on a small physical state space containing classical sectors and quantum matrix blocks. We quantify metastability by assuming that the evolution channel $\mathcal C$ changes little when applied a second time, with sufficiently small defect $\eta=\|\mathcal C^2-\mathcal C\|_\diamond$, and that the number of slow degrees of freedom is bounded independently of the full system size. Under these assumptions, we construct compression and reconstruction channels whose composition recovers every reduced state exactly, while the reverse composition gives an exactly idempotent channel approximating $\mathcal C$, answering Kitaev's exact-rounding question [Kit25]. Our constructions give an optimal diamond-norm bound of $\mathcal{O}(\eta^{1/3})$ on all input states, as well as an improved bound of $\mathcal{O}(\eta^{1/2})$ on metastable states prepared by $\mathcal C$, with constants depending only on the slow dimension. The reduced transition channel can be iterated to predict the microscopic dynamics with controlled error. We illustrate the QMSM through a weakly driven dissipative spin chain supporting either a metastable logical qubit or long-lived classical phases.

In this note, we study complete and quantum Gromov-Hausdorff convergence of operator systems obtained using smoothings of C*-algebras, where our smoothings arise from the action of a compact quantum group on the C*-algebra. Specifically, we show how if $\mathbb{G}$ is a co-amenable compact quantum group acting on unital C*-algebra $\mathcal{A}$, and both are equipped with suitable quantum metrics, bounded approximate identities in $L^{1}(\mathbb{G})$ may be used for approximations of the original space $\mathcal{A}$ under the complete and quantum Gromov-Hausdorff distance. We finish by presenting some applications to the convergence of spectral truncations of compact groups and connections to previous results.

A symplectic spectral function of a $2n \times 2n$ real positive definite matrix $A$ is a function of its symplectic eigenvalues $0< d_1(A) \leq \cdots \leq d_n(A)$, which is given by a composition $f \circ d$ of some symmetric function $f$ on the set of $n$-vectors with positive entries and the symplectic eigenvalue vector map $d(A)=(d_1(A),\ldots, d_n(A))$. In this work, we rigorously study various types of differentiability and Clarke generalized gradient of symplectic spectral functions, and also provide several applications of our findings. We show that the symplectic spectral function $f \circ d$ is Fréchet differentiable at $A$ if and only if $f$ is Fréchet differentiable at $d(A)$, and we compute the derivative expression explicitly. We also show that $f \circ d$ is strictly Fréchet differentiable (respectively, continuously Gâteaux differentiable) at $A$ if and only if $f$ is strictly Fréchet differentiable (respectively, continuously Gâteaux differentiable) at $d(A)$. We determine the Clarke generalized gradient of a symplectic spectral function $f \circ d$ at $A$, which is given in terms of the Clarke generalized gradient of $f$ at $d(A)$. As an application of our work, we show that the purity, von Neumann entropy, and Rényi entropy of a bosonic faithful Gaussian state are Fréchet differentiable functions of the covariance matrix of the Gaussian state. We also provide explicit expressions of their Fréchet derivatives.

We compute $K$-theoretic invariants for four three-dimensional operator systems to illustrate the general theory. We find that in some cases the invariants distinguish the operator system from its $C^*$-envelope, while in others they do not. We also explore extensions of the theory.

We compute $K$-theoretic invariants for several finite-dimensional operator systems. These include the Toeplitz operator system and its dual (the Fejér-Riesz operator system), as well as operator systems associated to tolerance relations, aka graph operator systems. A variety of techniques is applied, adapted to each of these cases, ranging from a generalization of Carathéodory's factorization result for block Toeplitz matrices, to Wiener-Hopf factorization and Schur complements. In most instances we find that the invariants coincide with their analogues for the $C^*$-envelope, except for the $K_1$-invariants of the Fejér-Riesz operator system.

Both the uniform Roe algebras and the quasi-local algebras encode the large scale geometry of metric spaces. Whether these algebras coincide is a question which goes back to Roe and has only been recently solved by Ozawa. We propose a dynamical point of view on this problem. Given a set $X$, every map $h\colon X\to\mathbb{R}$ induces a one-parameter group $\sigma_h$ of automorphisms of $\mathcal{B}(\ell_2(X))$, given by conjugation by the diagonal unitaries $e^{ith}$, and we show that the operators which are continuity points of $\sigma_h$ are precisely the uniform Roe algebra of the pseudo-metric induced by $h$. If we moreover assume that $X$ is a uniformly locally finite metric space and $h$ is allowed to range over all coarse maps, this gives dynamical characterizations of both algebras: $\mathrm{C}^*_{ql}(X)$ consists exactly of the operators which are continuous for all such flows, while $\mathrm{C}^*_u(X)$ consists exactly of norm limits of those which are analytic. The regularity conditions lying between continuity and analyticity then give rise to $\mathrm{C}^*$-algebras between $\mathrm{C}^*_u(X)$ and $\mathrm{C}^*_{ql}(X)$ which are invariant under bijective coarse equivalence, and we show that this scale is not degenerate: if $X$ is a coarse disjoint union of expander graphs, the algebra generated by operators which are analytic on a strip for every diagonal flow sits strictly between $\mathrm{C}^*_u(X)$ and $\mathrm{C}^*_{ql}(X)$.

We call an $n$-tuple of $2 \times 2$ matrices a “\textit{$2 \times 2$ matrix $\Gamma_n$-contraction}” if its joint eigenvalues (joint spectrum for a commuting tuple of matrices) are contained in $\Gamma_n$. In this short note on $\Gamma_n$-contraction, we produce two families of $2 \times 2$ matrix $\Gamma_n$-contractions whose fundamental operators do not satisfy the commutativity conditions and \begin{equation*} \begin{aligned} F_iF^*_{n-j} - F_jF^*_{n-i} = F^*_{n-j}F_i - F^*_{n-i}F_j, \quad 1 \le i, j \le n-1, \end{aligned} \end{equation*} which shows that the commutativity as well as the conditions on fundamental operators mentioned above are sufficient, but not necessary, for the existence of $\Gamma_n$-isometric dilation.

For Hermitian matrices $A$ and $B$, we prove that $(A\join B)\oplus(A\meet B)$ is unitarily equivalent to $A\oplus B$, where the join and meet are taken in Olson's spectral order. Taking traces answers a question of Bourin and Lee: for positive semidefinite matrices, $\Tr(A\join B)=\Tr(A+B)$ if and only if $A\meet B=0$. Iterating the direct-sum identity gives a trace formula for a finite family of positive matrices. The trace of the spectral supremum equals the trace of the sum precisely when the ranges form an algebraic direct sum. For each fixed $1<p<\infty$, the spectral supremum and the sum have equal Schatten $p$-norms if and only if the ranges are pairwise orthogonal. Finally we establish an elegant formula for the Frobenius inner product and the spectral order.

The optimal Sobolev regularity of the roots of a smooth curve of monic complex polynomials is $W^{1,q}$, for $q \in [1,\frac{d}{d-1})$, where $d$ is the degree. This result is stable in the sense that the solution map from $C^d$ coefficients to $W^{1,q}$ roots is continuous. We prove that, after a natural Lipschitz reparameterization, the roots are Lipschitz and the map from $C^d$ coefficients to reparameterization and reparameterized roots is continuous with respect to the $W^{1,q}$ topology on the target spaces, for all $q \in [1,\infty)$. The result is based on a convergence and reparameterization theorem for Almgren's $Q$-valued Sobolev functions.