Papers

Let $A, T \in M_n(\mathbb{C})$ and let $\Delta_A(T) = AT - TA$ denote the inner derivation induced by $A$. We determine when $T$ and $\Delta_A(T)$ are nilpotent under the second-order relation $$ \Delta_A^2(T) + \alpha\, \Delta_A(T) + \beta\, T = 0, \qquad \alpha, \beta \in \mathbb{R}, $$ according to the location of the roots of $z^2 + \alpha z + \beta$. If the roots have nonzero real parts of the same sign, then both $T$ and $\Delta_A(T)$ are nilpotent. If the roots are purely imaginary and nonzero, no nilpotency conclusion holds in general, whereas if one root is zero and the other is nonzero, $\Delta_A(T)$ is nilpotent but $T$ need not be. The double zero root gives the Kleinecke–Shirokov theorem. For real roots of opposite signs, the answer is governed by an arithmetic threshold: writing $z_1/z_2=-p/q$ in lowest terms, every solution is nilpotent when $p+q>n$, while for $p+q\le n$ an explicit cyclic construction yields solutions for which both $T$ and $\Delta_A(T)$ are invertible. Finally, for a relation $$ \sum_{k=0}^{m} c_k\,\Delta_A^k(T)=0 $$ of arbitrary order, every iterated commutator $\Delta_A^j(T)$ is nilpotent whenever the roots of the associated polynomial lie in an open half-plane whose boundary passes through the origin.

Let $\mathcal{H}$ be a complex Hilbert space and $\rho>0$. We establish the explicit formula \[ \omega_\rho (a\otimes b)=\frac{1}{\rho}\|a\| \|b\|+\left|1-\frac{1}{\rho}\right||\langle a,b\rangle|,\qquad a,b\in\mathcal{H}, \] for the $\rho$-numerical radius of rank-one operators, a unified expression interpolating between the operator norm, the numerical radius, and the spectral radius, which correspond to $\rho=1,2$, and $\rho\to\infty$, respectively. As an application, we derive a parametrized family of Buzano-type inequalities for four vectors. We extract from it an explicit closed-form bound and, under a reality condition which is automatically satisfied in real Hilbert spaces, we determine the best bound in the family in closed form. Examples show that the resulting bounds can be strictly sharper than both the Cauchy–Schwarz and the Buzano inequality, and the classical Buzano inequality is recovered as a boundary case, which yields an alternative proof of it.

Bayart, Pellegrino, and Rueda obtained diagonal absolute summability exponents for continuous \(m\)-linear operators from \((\ell_1)^m\) into \(\ell_p\). Their optimality was known for \(2\le p\le\infty\), whereas the case \(1\le p<2\) remained open. We settle the latter case by means of finite Walsh test operators, and consequently obtain the optimal exponent for every \(1\le p\le\infty\). The same argument, combined with the corresponding interpolation and embedding results, also yields optimal Kwapień-type statements for several related range spaces.

We characterize exact nonuniqueness in the planar interior Radon problem for arbitrary pairs of open convex sets. There exists a nonzero smooth function compactly supported in the first set whose integral over every line meeting the second set vanishes if and only if an ellipse contains the closure of the second set and is compactly contained in the first set. The same criterion holds for a nonzero $L^1$ function whose essential support is compactly contained in the first set and whose Radon transform vanishes almost everywhere on those lines. For concentric open squares, the criterion gives the sharp threshold $1/\sqrt2$ for the inner-to-outer half-side ratio. This yields counterexamples to Conjecture 1.2 of Boman (2021) and to Theorem 40.1 of Boman (2025).

Motivated by the concept of numerical hypercyclicity, in this paper, we introduce three new notions in linear dynamics: topological numerical transitivity, generalized numerical transitivity, C*-transitivity. The first notion is defined for operators on general Banach spaces, whereas the latter two are formulated in the setting of operators on Banach algebras. The concept of C*-transitivity is also applicable to operators on the space of Hilbert-Schmidt operators. We prove that these new notions are mutually different, and we also show that they differ from the standard notions in dynamics. In particular, while topological numerical transitivity implies numerical hypercyclicity, as we argue in the paper, we provide examples of numerically hypercyclic and strongly numerically hypercyclic operators that are not topologically numerically transitive. Also, we construct nonsupercyclic C* transitive and generalized numerically transitive operators on the C*-algebra of compact operators on a separable Hilbert space, as well as C*-transitive nonsupercyclic operators on the standard Hilbert C*-module. In addition, we study diagonal operators on finite-dimensional and separable Hilbert spaces, and we obtain complete characterizations of both numerical hypercyclicity and topological numerical transitivity in terms of coefficient-simplex criteria. Further, we provide some sufficient conditions for diagonal operators on the standard Hilbert C*-module to be C*-transitive. All these results are illustrated with concrete examples. At the end of the paper, we compare topological numerical transitivity and C*-transitivity with numerous standard notions in linear dynamics, and we prove that topological numerical transitivity and C*-transitivity genuinely differ from all these standard concepts in dynamics.

We prove a sharp higher-order Rademacher-chaos inequality on the Hamming cube using random-reveal martingales. As a consequence, we obtain the sharp metric $X_p$ inequality for $L^p$, resolving an open question posed by Naor.

In this paper, we study various results on numerical range of weighted composition operators on Hardy space of Dirichlet series. Certain conditions are provided under which the numerical range contains zero and some circular or elliptic disks. With the help of invariant subspaces of the operator, we also determine the numerical range of reductive weighted composition operators on this space. Additionally, we discuss some results on the Davis-Wielandt shell of composition operators with some natural questions that arise from our findings.

We characterize the ambiguity in reconstructing bounded measurable sets from their Fourier transforms sampled on a lattice. For a full-rank lattice \(\mathcal L\subset\R^d\), we prove that two bounded measurable sets have identical indicator Fourier transforms on \(\mathcal L^*\) if and only if they are equidecomposable, modulo null sets, into finitely many measurable pieces by translations in \(\mathcal L\). If the sets are connected finite unions of convex bodies and no two distinct points of either set are congruent modulo \(\mathcal L\), we show that this equidecomposition reduces to a single translation by a vector of \(\mathcal L\). In particular, two such sets that are symmetric about the origin must coincide. We also construct counterexamples in dimensions two and three to a conjecture of the second author, which had a similar hypothesis, but lacked the anti-aliasing hypothesis. Together with the first author's counterexamples in dimensions at least four, these show that the conjecture fails in every dimension $d\geq 2$. The positive results here are an outgrowth of refinements of this conjecture.

In this paper, we find the explicit representation of the set-valued metric projection operator from l1 to the unit closed ball in l1. By using this representation, we investigate the properties of the Mordukhovich derivatives of the set-valued metric projection operator, which is applied to calculate its covering constant. As a special case, we consider the 2-d Banach space. We find the explicit solutions of the set-valued metric projection operator from the considered 2-d Banach space to its unit closed ball. By these solutions, we will calculate its Mordukhovich derivatives in details.

We study the dependence of the transfinite Schreier hierarchy on the choice of fundamental sequences for the countable limit ordinals. With each Schreier family we associate an interval endpoint function. We prove that the bounding number is equal to $\omega_1$ exactly when the fundamental sequences may be chosen so that these endpoint functions form an unbounded family in the eventual domination order. Equivalently, the hierarchy then satisfies the tail-covering property isolated by Shiliaev, or every infinite compact interval family has infinite intersection with some member of the hierarchy. We also prove that the dominating number is equal to $\omega_1$ exactly when the fundamental sequences may be chosen so that every compact family of finite subsets of $\{2,3,\ldots\}$ is contained in one Schreier family. The latter equivalence combines Fremlin's cofinality theorem for compact subsets of the rationals with an absorption construction for compact families.

We develop a novel approach to matrix discrepancy based on matrix small-ball estimates. Specifically, we use a determinantal weight (obtained from the log-barrier) to scale the small-ball probability into a partition function of a tilt of the Gaussian measure. We then employ matrix-weighted Poincaré inequalities to compare this partition function to that of a pinched or diagonal part of the matrix, obtaining dimension-free constants. Our technique yields a hereditary small-ball estimate for Gaussian series that should be of independent interest. As the main application, we resolve the Matrix Spencer conjecture: for symmetric $n\times n$ matrices $A_1,\dots,A_n$ with $\|A_i\|\le1$, one can efficiently find a coloring $x\in\{\pm1\}^n$ with $\|\sum_{i=1}^n x_iA_i\|=O(\sqrt n)$.

We prove that the boundary of every bounded $(M^{1,p},M^{1,q})$-extension domain in a $Q$-doubling metric measure space has measure zero whenever $0<q<p<\infty$, with the additional restriction that $p<qQ/(Q-q)$ if $q<Q$.

We introduce a notion of continuity equation on metric spaces that is capable of describing curves of probability measures which are absolutely continuous, and more generally of bounded variation (BV), with respect to the 1-Wasserstein distance. This continuity equation is based on a notion of measure-valued derivations, whose basic theory is also developed in this paper. On $\mathbb{R}^n$, our formulation is consistent with the continuity equation with singular flux introduced by Almi–Rossi–Savaré (arXiv:2506.15333), including the corresponding notion of minimal solutions. In this work, we characterize BV-curves in the space of probability measures equipped with the (extended) 1-Wasserstein distance as those curves satisfying the continuity equation with a measure-valued derivation of finite mass. To this aim, we extend our previous work (Calc.Var.(2024)63:16) on probabilistic representations on BV-curves and construct from them measure-valued derivations (resp. flux measures) on geodesic metric spaces (resp. on $\mathbb{R}^n$).

For finite-dimensional linear inverse problems where the variables are Gaussian, it is well-known that the minimum-mean-square error estimator takes the form of a regularized least-squares data fit. In this chapter, we show that this equivalence extends to a much broader infinite-dimensional setting where generalized splines take the role of linear regressors and generalized Gaussian processes on a nuclear space $S$ are the counterpart of Gaussian random vectors. The scope of this extension is of the same nature as the switch from the classic notion of function to that of a distribution, also known as a "generalized function." Our formalism involves a whitening/regularization operator $L: S\to S'$ whose continuous extension induces a native Hilbert space $H\subset S'$ that plays a central role in our characterization. The presentation is self-contained for the most part and remarkably general and powerful. It allows for the recovery of all known instances of such equivalences; in particular, the methods involving innovations and reproducing-kernel Hilbert spaces developed by Kailath and his students, and the mathematical correspondence between fractional splines and Mandelbrot's fractional Brownian motion (fractals), with the former being the optimal estimators of the latter. It also covers general Bayesian methods for the resolution of infinite-dimensional inverse problems.

Given a pair of commuting, ergodic, trace-preserving and trace-integrable actions of locally compact unimodular groups on a common semifinite von Neumann algebra we prove a form of associativity for convolutions of triples of operators. We view this as a basis for a general version of quantum harmonic analysis.

We show that length functions on hyperbolic groups whose induced metrics are hyperbolic and weakly geodesic naturally give rise to compact quantum metric spaces, thus generalizing the result of Ozawa and Rieffel. As a result we obtain several new examples of compact quantum metric spaces, in particular arising from Green metrics associated with random walks on hyperbolic groups, and from proper cocompact isometric actions on proper geodesic hyperbolic spaces admitting a point with trivial stabilizer.

We solve the matricial univariate rational truncated moment problem on the real line and on a half-line. We give explicit necessary and sufficient conditions for the existence of a positive matrix-valued representing measure and show that every solvable problem admits a finitely atomic representing measure whose total atomic multiplicity equals the rank of the associated moment matrix. The rational problem is reduced to an ordinary matricial moment problem with the additional requirement that the representing measure avoid the real poles of the rational data. The main new ingredient is a simultaneous prescribed-node result: the smallest attainable multiplicities at finitely many prescribed points can be realized by a single minimal representing measure. Our proofs are constructive and use flat extensions, block-column relations, and localizing conditions.

We study finite-fibre determinants for rational Bishop operators and their role in cyclicity for irrational parameters. The paper has two main parts. First, for the constant vector $f=1$, a resultant identity and a discrete Fourier factorization reveal a determinant parity mechanism for general modular orbit order: odd denominators give a nonnegative normalized determinant on the positive fundamental cell, while for even denominators the unique real alternating Fourier mode is the only factor capable of producing a sign-changing zero. We give an explicit example at $(r,q)=(9,16)$ and an analytic infinite family $(r,q)=(3,6n-2)$. Grivaux's zero-free determinant is identified as the consecutive-order subfamily $D_{1,q}$, so these zeros are caused specifically by nonconsecutive modular ordering. Second, we prove an explicit cyclicity criterion that does not require global nondegeneracy or monotonicity of the fibre determinant. A quantitative Remez estimate controls the small-determinant set; cutoff inverses are approximated by endpoint-corrected Fejer polynomials; and an explicit continuity modulus transfers the resulting rational approximants to irrational parameters. This produces a fully explicit continued-fraction gap function for $f=1$ and, more generally, for every polynomial $f$ with $f(0)\ne 0$. The argument also gives the exact degree and leading coefficient of the corresponding polynomial-vector fibre determinants.

We study $\ell_0$ isoperimetry for a convex body $K\subset \mathbb{R}^n$, $n\ge2$. For a Borel set $S\subset K$, let $\partial_0^K S$ be the set of points in $K \setminus S$ that can be reached from $S$ by changing at most one coordinate (i.e. the $\ell_0$ boundary of $S$). Suppose that, for some unconditional convex body $Q \subset \mathbb{R}^n$, numbers $r,R>0$, and possibly different centers $x_0,y_0$, \[ x_0+rQ \subset K\subset y_0+RQ. \] Writing $s=\text{vol}(S)/\text{vol}(K)$, we prove that whenever $0<s \le 1/2$, \[ \frac{\text{vol}(\partial_0^K S)}{\text{vol}(S)} \ge \frac{cr}{nR} \min\left\{1,\frac{\log(e/s)}{n}\right\}, \] where $c > 0$ is an absolute constant. Consequently, the associated $\ell_0$-isoperimetric coefficient is at least $cr/(n^2R)$. Previous direct lower bounds were only known for $\ell_2$ and $\ell_\infty$ regularity whereas our lower bound holds directly for any $Q$-regularity, where $Q$ is an unconditional convex body. Compared to $\ell_2$ and $\ell_\infty$ regularity, our lower bound result improves upon the previously best known lower bounds, for any $s$, by a factor of $n$. As an application of our result, we give improved mixing time bounds for the Coordinate Hit and Run walk (CHAR). Our proof of the lower bound is based on a modification of the method of canonical paths applied to a continuous Hamming graph over our convex body. Our construction of canonical paths can be viewed as a suitable coordinate discretization of certain mass transport maps from $S$ to $S^c$. We also give complementary upper-bounds for any $Q$-regularity, with an overall factor of $n$ gap between the two.

We establish a version of the classical results concerning descriptions of isometries on $C^*$-algebras and noncommutative $L_p$-spaces due to Kadison (1951) and Yeadon (1981) in the setting of Haagerup–Schultz algebras. Precisely, we show that (not necessarily surjective) isometries on such algebras are necessarily implemented by partial isometries and trace-preserving Jordan $^*$-monomorphisms.

We study the bifurcation of two-dimensional Darcy-Bénard convection (DBC) in a rectangular domain, a canonical model for thermal convection in porous media with applications in geophysics and engineering. The momentum equation lacks advection and viscous dissipation, being regularized solely by a linear Darcy damping term. As a result, the linearized operator generates a semigroup that is neither analytic nor compact and the nonlinear term fails to be Lipschitz. The system is thus placed outside the scope of the standard center-manifold theorem. To overcome these obstructions, we develop a center-manifold reduction adapted to DBC system. Our main result is a constructive proof of the existence of the center manifold function h and local attractivity of the center manifold-the exponential convergence of small solutions toward it-without relying on analyticity of the full linear semigroup and Lipschitz nonlinearity. We circumvent these difficulties by exploiting the partially dissipative structure: the temperature equation is governed by the Laplacian, which generates an analytic semigroup and provides the smoothing needed to compensate for the lack of regularity in the velocity and the absence of global Lipschitz bounds. Through carefully designed inequalities, we establish both the construction of the center manifold function h and the exponential convergence of nearby solutions. Explicit approximations for the center manifold are derived in two scenarios-one simple eigenvalue and two distinct eigenvalues-yielding reduced systems of ordinary differential equations whose analysis determines the bifurcation type. Numerical simulations are presented to corroborate the theoretical results.

Let $\mathbb{P}_n$ be the cone of $n\times n$ complex positive definite matrices. For $0<\theta<1$ and $r\ge0$, let $M_r^{(\theta)}$ denote the weighted matrix power mean, with the logarithmic endpoint at $r=0$, defined by $$ M_r^{(\theta)}(A,B) = \begin{cases} \big((1-\theta)A^r+\theta B^r\big)^{1/r}, & r>0,\\[2mm] \exp\big((1-\theta)\log A+\theta\log B\big), & r=0. \end{cases} $$ For $p>q\ge0$, we introduce $$ \Phi_{p,q}^{(\theta)}(A,B) = \operatorname{Tr}\big[M_p^{(\theta)}(A,B)-M_q^{(\theta)}(A,B)\big]. $$ We prove that $\Phi_{p,q}^{(\theta)}$ is a quantum divergence in the sense of Bhatia-Gaubert-Jain. Moreover, for $$ 0\le q\le1\le p\le2,\qquad p>q, $$ it is jointly convex and satisfies the data processing inequality for completely positive trace-preserving maps, with the output side interpreted by the regularized boundary convention whenever it is singular. The construction recovers Hellinger-type and Jensen-Shannon-type quantities as special cases or limits. We also compute the induced local quadratic form and provide scalar and matrix obstructions showing that the convexity and data-processing range cannot be extended to all parameters.

We study linear cocycles over ergodic translations of the two-torus acting on Hermitian vector bundles described by unitary sewing matrices. For a continuous invariant complex line subbundle $L$, we prove that $c_1(L) \ne 0$ obstructs every nonzero measurable eigensection under a measurable phase-coboundary condition. The abstract obstruction holds in every rank, while for rank-two analytic cocycles with a dominated splitting over a Diophantine translation the phase condition is automatic. In that regime the criterion is exact: $L$ carries a nonzero measurable eigensection if and only if $c_1(L)=0$ and the winding vector of the multiplier phase vanishes. When these conditions hold, the realized eigenvalues form a dense coset on a circle; for each such eigenvalue, the eigenspace carried by $L$ is one-dimensional and has a real-analytic nowhere-vanishing generator. We also construct a Liouville example showing that the Diophantine hypothesis cannot be removed. The proof converts the Chern number into a magnetic charge on the universal cover and applies covariance for irrational magnetic translations.

We characterize all lattices $\Lambda \subset \mathbb{R}^2$ of rational density for which the Gabor system generated by the first Hermite function along $\Lambda$ is a frame for $L^2(\mathbb{R})$. We prove that these are precisely the lattices whose density satisfies $D(\Lambda) = \frac{q}{p} $ where $p,q \in \mathbb{N}$ are coprime with $q \geq p+2$, thereby confirming a conjecture of Lyubarskii and Nes. A formalization of our main result in Lean $4$ is also included.

We establish a noncommutative version of a familiar Johnson–Maurey–Schechtman–Tzafriri Theorem, by showing that for any $0 <p<2$ and a (not necessarily semifinite) von Neumann algebra $\mathcal{M}$ on a separable Hilbert space, if a symmetric quasi-Banach function space $E(0,1) $ containing the function $t\mapsto t^{-1/p}$, $0<t\le 1, $ then there exists a noncommutative probability space $(\mathcal{N},\sigma )$ such that $L_p(\mathcal{M})$ is isometric to a subspace of $E(\mathcal{N},\sigma)$. In particular, this answers two questions raised by Randrianantoanina in 2006.

In the framework of network dynamics, learning models, and neural tangent kernels (NTK), we show that the corresponding linearized dynamics leads naturally to a semigroup formulation. More precisely, in our analysis of input/output models, the time-dynamics is presented via special semigroups of linear operators on Hilbert spaces, together with an associated class of semigroup perturbations. In this context, we then present new and explicit a priori perturbation-bound results: for the fixed-kernel linearization constructions arising in the NTK setting, we prove norm-bounds on the corresponding semigroup perturbations, in the form of explicit finite-time perturbation estimates. We further present refinements on prescribed task spaces, Cesàro-averaged (ergodic) comparisons estimates, and versions in which the lower spectral edge assumption is replaced by a spectral-distribution condition. We also extend the comparison to nonautonomous NTK evolutions through piecewise-frozen approximations, record a corresponding discrete Euler specialization, and offer worked examples in order to illustrate our perturbation-bound estimates.

In this paper, we study Lipschitz spaces adapted to Dunkl–Schrödinger operators $\mathcal{L} = -\Delta_k + V$, where $\Delta_k$ is the Dunkl Laplacian and $V \ge 0$ belongs to the reverse Hölder class $\mathrm{RH}^q(dw)$ with $q > \max(1, \mathbf{N}/2)$. For $0 < \beta < 2$, we establish the full equivalence between the heat semigroup-based Lipschitz spaces $\widetilde{\Lambda}_{\mathcal{L},k}^{\beta/2}(\mathbb{R}^N)$ and the pointwise weighted Zygmund classes $\Lambda_{\mathcal{L},k}^{\beta}(\mathbb{R}^N)$ defined via the critical radius function $m(\mathbf{x})$. Remarkably, this characterization shows that the spatial smoothness component decouples completely from the underlying reflection group symmetries and root systems. As an intermediate result, we establish new equivalences for inhomogeneous weighted Lipschitz spaces associated with the unperturbed Dunkl Laplacian $\Delta_k$. A central difficulty in this context — arising from the necessity to disentangle the Euclidean distance from the orbit metric $d(\mathbf{x},\mathbf{y})$ — is resolved through a detailed geometric analysis. Finally, we provide pointwise Gaussian upper bounds for the time derivatives of the Dunkl–Schrödinger heat kernel, which are of independent interest for future work in harmonic analysis on Dunkl structures. Crucially, we do not assume the potential $V$ to be $G$-invariant under the action of the Weyl group, which introduces major geometric difficulties resolved here via a local analysis of the orbit metric.

Jiang, Lee, and Vempala conjectured that if $X,Y\in\mathbb{R}^n$ are independent isotropic log-concave random vectors, then $W_2(L(\langle X,Y\rangle),N(0,n))$ is bounded by a universal constant. Subject to Theorems 1.2 and 2.5 of arXiv:2607.24164v1, we prove this conjecture and a rectangular bilinear-form extension. For independent isotropic log-concave $X\in\mathbb{R}^m$, $Y\in\mathbb{R}^n$, and nonzero $B\in\mathbb{R}^{m\times n}$, put \[ r_4(B)=\frac{(\operatorname{Tr}(B^\top B))^2} {\operatorname{Tr}((B^\top B)^2)}. \] We construct a nonnegative scalar Stein kernel for $X^\top B Y/\|B\|_F$ whose squared $L^2$ discrepancy is at most $20/r_4(B)$, and consequently obtain the same bound for squared $2$-Wasserstein distance to $N(0,1)$. The proof develops an exact covariance identity and deficit decomposition for trace observables of moment-map Stein kernels, together with a stability theorem for positive Stein kernels under log-concave approximation. Taking $B=I_n$ yields \[ W_2^2\left(L\left(\frac{\langle X,Y\rangle}{\sqrt{n}}\right),N(0,1)\right)\leq\frac{20}{n}, \] which is the Jiang–Lee–Vempala conjecture.

We study Birkhoff-James orthogonality and local symmetric points in the space $C(K,X)$ of vector-valued continuous functions equipped with a numerical-radius type (semi-)norm. We extend the theory of abstract numerical ranges to semi-normed spaces and obtain complete characterizations of the left and right symmetric points of $C(K,X)$ with respect to the numerical-radius (semi-) norm. We also establish corresponding results for the space $\mathcal{K}(X)$ of compact operators on $X$.

We settle the complete Crouzeix conjecture for matrices of order at most three and prove the Q-algebra case of the completely bounded Clouâtre–Ostermann–Ransford (COR) conjecture for homomorphisms into matrices of order at most three, via sharp abstract column and row estimates. Our approach also establishes the scalar COR conjecture in a stronger form, for homomorphisms with commutative range on Banach algebras with unity satisfying von Neumann's inequality. Under contractivity of the symmetrized map, this result holds on arbitrary Hilbert spaces without initial boundedness assumptions on the homomorphism or the antilinear map. For arbitrary operator algebras, we disprove the complete COR conjecture by an exact three-dimensional example with target matrix order two. We also prove the sharp complete bound for every matrix subalgebra containing the diagonal, in arbitrary matrix order and on arbitrary target Hilbert spaces. We also obtain column and row square-function inequalities with sharp norm bounds, strict scalar bounds for operators similar to normal operators, sharp complete spectral constants for scaled $q$-numerical ranges in dimensions two and three, and rigidity, stability, and representing-measure results.