Papers

In this article we study the antitonicity property of the Moore-Penrose inverse in the class of selfadjoint operators, with respect to the Löwner order. For this purpose, we employ different positive decompositions that selfadjoint operators admit. In addition, we relate a weak version of the antitonicity property with Thompson components.

Let $0<p<q\leq 1$. We show that the Schur $p$-property provides a powerful structural principle for Lipschitz-free $p$-spaces. Our main result asserts that $\mathcal{F}_p(M)$ has the Schur $p$-property for every $q$-metric space $M$; when $M$ is compact, it has the strong Schur $p$-property, with a constant depending only on $p$ and $q$. As consequences, $\mathcal{F}_p(M)$ is $\ell_p$-saturated, contains no isomorphic copy of an infinite-dimensional $r$-Banach space for $p<r\leq 1$, and every bounded operator from a $p$-Banach space into $\mathcal{F}_p(M)$ is either compact or fixes a copy of $\ell_p$. These results settle Questions 6.1, 6.2, 6.5, and 6.6 from our recent work [F. Albiac, J. L. Ansorena, J. Bíma and M. Cúth, Lipschitz free p-spaces for $0<p<1$ in the light of the Schur $p$-property and the compact reduction, J. Geom. Anal. 36 (2026), Paper No. 54] on the Schur $p$-property, together with several related problems. They also confirm a prediction of Kalton and the first-named author from 2009: no infinite-dimensional $q$-Banach space has the $p$-Lipschitz lifting property. Further applications reveal a sharp contrast between the linear and Lipschitz structures of nonlocally convex spaces.

For any convex set $\Omega\subset\mathbb{R}^n$ that does not contain affine lines, we prove the inequality $$\int_\Omega\frac{|\hat{f}(x)|^2}{\omega_\Omega(x)}\,dx\leq C(n)\|f\|_{L^1}^2,\quad \supp\hat{f}\subset\Omega,$$ where $\omega_\Omega(x)=m(\Omega\cap(2x-\Omega))$. As a consequence, we derive a weak factorization for $$\PW^1(\Omega)=\{f\in L^1(\mathbb{R}^n):\supp\hat{f}\subset\Omega\}.$$ Furthermore, we establish a complete characterization of Schatten class Hankel operators for polyhedra for all $1\leq p<\infty,$ extending the already known $1\leq p\leq 2$ range.

In this paper, we will investigate the connection between Frechet derivatives and Mordukhovich derivatives of single-valued mappings in Banach spaces. We will find some properties of Mordukhovich derivatives of set-valued mappings, which will be demonstrated by the set-valued metric projection operator in finite dimensional Banach spaces. We introduce the concept of locally variational inequalities in Banach spaces, which focus on finding solutions within some specific neighborhoods rather than across the entire global domain; and we find the connection between the solutions of locally variational inequality problems and the Mordukhovich derivatives.

In this paper, we characterize generalized tri-circular projections on the following three vector-valued spaces of analytic functions on the open unit disk $\mathbb{D}$: the Hardy space $H^p(\mathcal{K})$ for a complex separable Hilbert space $\mathcal{K}$ and $1 \le p < \infty,p \neq 2$; the space $H^\infty(E)$ of $E$-valued bounded analytic functions for a Banach space $E$ with trivial multiplier algebra; and the little Bloch spaces $\mathcal{B}_0(\mathbb{D}, E)$ and $\mathcal{B}_{\ast}(\mathbb{D}, E)$, for $E$ smooth, strictly convex, and reflexive.

We present a new method to prove uncertainty principles for functions of low energy in a very general set-up. This leads to results for metric measure spaces, Dirichlet spaces and graphs under rather weak assumptions.

Let $\mathcal{H}$ be a complex Hilbert space. We characterize the bijective linear maps $\Phi:\mathcal{B}(\mathcal{H})\to\mathcal{B}(\mathcal{H})$ that preserve the spectral radius, respectively the numerical radius, on pairs of unitarily similar operators. We also characterize bijective linear maps that transform unitary similarity into similarity. Our results cover both finite and infinite-dimensional Hilbert spaces, with no separability assumption in the latter case.

We construct counterexamples to Rockafellar's sum conjecture in which two maximally monotone operators satisfy the interior-domain condition but their sum is not maximally monotone, thereby providing the complete disproof of the conjecture. We establish a general construction theorem that computes the entire monotone polar of a class of graphs and characterizes their maximal monotonicity by the nonexistence of solutions to explicit equations in the continuous dual. We also prove a pullback theorem that transfers counterexamples through bounded linear surjections. These theorems provide a systematic mechanism for generating entire families of counterexamples and lead to further structural consequences for the resulting operators. Specifically, we obtain four classes of counterexample families: weighted constructions with different curves, first operators with prescribed affine value dimensions, second operators obtained by positive rescaling and norm-continuous monotone perturbation with full domain, and counterexamples on further Banach spaces. The last class yields counterexamples on every Banach space containing a closed subspace isomorphic to $c_0$ or $\ell^1$, or admitting such a quotient. We also give explicit constructions on $c_0$ and standard $\ell^1$ that realize the construction and pullback mechanisms, respectively. We further determine the domain geometry and exact radial bounds of the constructed operators, characterize reflexivity by a fixed rank-one test in two classical classes of Banach spaces, identify maximal monotone extensions under surjective pullback, and compute exact Fitzpatrick identities. The appendices further extend these constructions to additional parameter and product families, nonlinear scalar and strictly monotone second operators, normal-cone and subdifferential partners, and examples with prescribed radial bounds, and more counterexample families.

We prove that the intrinsic jump set of every locally integrable function on an equiregular Carnot-Carathéodory space is intrinsically countably rectifiable. A key new ingredient is a local scalar Whitney extension theorem for intrinsic $C^1$ functions.

We address a structural gap in the quantitative theory of operator approximation by investigating how subspace approximation schemes interact with local reflexivity. Recent literature establishing the duality of scheme-relative approximation numbers, $a_n(T,Q) = a_n(T^{**}, Q^{\perp\perp})$, has relied on imposing an Extended Local Reflexivity Property (ELRP) as an independent axiom. In this note, we establish a dichotomy. For infinite-dimensional admissible schemes, we prove they naturally generate complete nests, perfectly satisfying the topological hypotheses of the Oja-Veidenberg Nest Principle of Local Reflexivity. Conversely, we demonstrate that for approximation schemes with finite-dimensional components, the bidual geometry collapses. Consequently, the ELRP is entirely superfluous, and the duality of approximation numbers holds unconditionally for all bounded linear operators via an elementary isometric restriction.

We consider a broad class of exchange dynamics with arbitrary weighted graph or hypergraph update structures, which includes the Kipnis–Marchioro–Presutti (KMP) model and the energies of Kac's walk on the sphere. These are conservative continuous-spin systems whose reversible measures are Dirichlet distributions. We prove that their spectral gap is always attained by a polynomial of degree at most two in the energy variables. Equivalently, through an intertwining with the associated discrete particle systems, the dominant mode is always represented by either a one-particle or a two-particle observable. We interpret this as a manifestation of an Aldous-type spectral gap phenomenon. In particular, the resulting two-particle spectral gap identity settles a recent conjecture of Alon and Puder. We also characterize sharply when one particle suffices and when a genuinely two-particle mode dominates. Under a common rescaling of the Dirichlet parameters, segment-like geometries are precisely those for which the gap is always of one-particle type, mean-field geometries are precisely those for which it is always of two-particle type, and all other geometries exhibit a nontrivial transition. The proof relies on the analysis of the so-called hidden model, a dual representation of the original process. The mechanism underlying the degree-two reduction is remarkably robust and extends to a much broader class of stochastic exchange models and their associated particle systems, including the harmonic process, the immediate exchange model, averaging-type processes, and nonreversible variants. Finally, we analyze boundary-driven versions of the KMP model, in which the bulk exchange dynamics interacts with reservoirs, generally resulting in nonreversible processes. In contrast to the conservative setting, we prove that the spectral gap of a boundary-driven model is always of one-particle type.

It was proved in the 1980s by Burkholder and Bourgain that, for any Banach space $X$ and $1<p<\infty$, the UMD$_p$ property for $X$ is equivalent to boundedness of the Hilbert transform on $L^p(\R;X)$, and that the UMD constant $\beta_{p,X}$ and the Hilbert transform constant $\hbar_{p,X}$ are related by the quadratic bounds \begin{equation*} \hbar_{p,X}\lesssim(\beta_{p,X})^2, \qquad \beta_{p,X}\lesssim(\hbar_{p,X})^2. \end{equation*} In this paper we present examples showing that both bounds are sharp. More precisely, we construct explicit $2^n$-dimensional Banach spaces for which the Hilbert transform constant grows like $n$ and the UMD constant like $\sqrt n$, and a second family with the reverse behaviour.

In this article, we establish $\mathrm{L}^p$-mapping properties for the generalized Stokes operator with bounded measurable coefficients on the full range of exponents that is known to be sharp for elliptic systems with rough coefficients. These include $\mathrm{L}^p$-estimates for the corresponding semigroup, its gradient and the $\mathrm{H}^\infty$-calculus, as well as $\mathrm{L}^q$-$\mathrm{L}^p$ smoothing estimates. As a key ingredient, we develop a non-local framework for $\mathrm{L}^p$-extrapolation that captures the relationship between hypercontractivity, non-local decay estimates and uniform $\L^p$-bounds.

We prove the sharp Hausdorff–Young inequality on the quantum Euclidean space. Our result implies the sharp Hausdorff–Young constants for the Weyl transform, as well as that for Heisenberg groups. The key ingredient is a novel flow related to the mixed-norm of noncommutative Gabor transform. This, meanwhile, implies a new proof of the classical sharp Hausdorff–Young inequality. We then apply the sharp Hausdorff–Young inequality to establish the sharp Young inequality for noncommutative convolution in the range $1\le p,q\le2\le r\le\infty$, with $1/p+1/q=1+1/r$. After appropriate rescaling and trace normalization, this convolution coincides with beam-splitter convolution for bosonic systems, yielding the corresponding Young inequalities with optimal constants in the same range.

We study the density of smooth external forces for which the three-dimensional Navier–Stokes equations lose classical regularity by a prescribed time $T>0$. Starting from the compact forced blowup solution of OpenAI, we construct a blowup solution near every given smooth solution while preserving its initial velocity. A smooth cutoff of a local vector potential makes the given velocity vanish near the support of a rescaled blowup solution. The two velocities then have no nonlinear interaction, and the resulting force remains smooth through the blowup time. For fixed viscosity and zero initial velocity, such forces are dense in the relative $L^1_tH^s_x$ topology on both $\mathbb{T}^3$ and $\mathbb{R}^3$ exactly when $s<1/2$.

We extend Cuenin's compact-manifold spectral bounds for Schrödinger operators with complex potentials to a general pseudodifferential setting. More precisely, we study operators \(P+V\), where \(P\) is a positive self-adjoint elliptic classical pseudodifferential operator of positive order and \(V\) is complex-valued. The main analytic input is a resolvent principle showing that spectral cluster estimates for \(P\) imply \(L^p\)-\(L^{p'}\) resolvent estimates along suitable complex curves. Combined with Sogge's spectral cluster bounds, this yields exterior-region resolvent estimates extending those of Krupchyk and Uhlmann; we also prove direct resolvent bounds in the interior region. On Zoll manifolds, we discuss the sharpness of the resulting spectral bounds.

We study the dynamics of self-consistent transfer operators associated with mean-field coupled systems whose unit dynamics is given by an interval map with finitely many singularities of cusp type. We show that, when the coupling strength is sufficiently weak, such an operator has a unique fixed point in a suitable cone of the associated Banach space, and that this fixed point attracts the orbit of every density in the cone at an exponential rate in the corresponding norm.

In this paper, we study composition operators that act between different de Branges-Rovnyak spaces. Our main results are suggested by a paper of Mashreghi and Shabankhah concerning composition operators between model spaces. We also answer several related open questions posed by Dellepiane and Seco. To prove these results, we apply reproducing kernel Hilbert space methods, Sarason's approach to composition operators as integral operators, and Aleksandrov-Clark measures.

In a tracial von Neumann algebra M, two subsets X and Y are said to be N-independent if alternating products of length at most N of trace zero words from X and Y have trace zero. We show that two approximately 2N-independent finite tuples can be perturbed, in operator norm, to be exactly N-independent. Consequently, we are able to generalize Theorem 5.1 and Theorem H of Houdayer-Ioana and resolve Problem 31 of Kunnawalkam Elayavalli in the positive.

Let $g$ be a nondecreasing left-continuous function and let $\mu_g$ be its Lebesgue–Stieltjes measure. We establish a Banach-valued fundamental theorem of calculus and an Aubin–Lions compactness theorem for evolution measured by $\mu_g$, allowing absolutely continuous, singular continuous and atomic components. If the range space has the Radon–Nikodým property, a curve is $g$-absolutely continuous if and only if it is an indefinite Bochner integral; its strong $g$-derivative is the Bochner density and the variation measure has density equal to its norm. A terminal atom may make the derivative invisible from the Bochner state class, so the natural evolution space is a graph of state–derivative pairs. For $B_0\Subset B\hookrightarrow B_1$, with $B_0$ and $B_1$ reflexive and $1<p_0,p_1<\infty$, the state projection is a compact linear operator into $L_g^{p_0}([a,b);B)$. The proof combines bounded evaluation at atoms, local $\mu_g$-averages at nonatomic points and Ehrling's inequality. The result recovers the classical theorem and yields compactness principles for bounded time scales, weighted sequences and mixed Stieltjes measures.

Consider the Dirichlet-type spaces on the bidisc defined by $$\mathcal{D}_{\beta}(\mathbb{D}^2)=\Bigg\{f(z_1,z_2)=\sum_{k,l}a_{kl}z_1^kz_2^l\in\mathcal{O}(\mathbb{D}^2): \sum_{k,l\ge 0}|a_{kl}|^2(k+l+1)^{\beta}<+\infty\Bigg\}.$$ Given $\beta_c\in(0,2],$ we construct a function $f$ that belongs to the Dirichlet-type space $\mathcal D_{2}(\mathbb{D}^2)$ of the bidisk and is cyclic in $\mathcal D_\beta(\mathbb{D}^2)$ if and only if $\beta\leq \beta_{c}.$ We also show that the critical index satisfies $\beta_c=2-\mathrm{dim}_H(\mathcal{Z}(f)\cap \mathbb{T}^2),$ where $\mathrm{dim_H}(\mathcal{Z}(f)\cap \mathbb{T}^2)$ is the Hausdorff dimension of the zero set of the function $f$ on the two-torus $\mathbb{T}^2.$

We prove that every trace-zero matrix $A\in M_n(\mathbb{C})$ admits a representation $A=BC-CB$ with $B,C\in M_n(\mathbb{C})$ and $\lVert B\rVert\lVert C\rVert\le K\lVert A\rVert$, where $K$ is an absolute constant independent of $n$, and $\lVert\cdot\rVert$ denotes the operator norm. For a fixed $t>0$, the proof splits according to whether $\lVert\operatorname{Re}(e^{\mathrm{i}\theta}A)\rVert_1\ge tn\lVert A\rVert$ holds for all $\theta\in\mathbb{R}$, where $\lVert\cdot\rVert_1$ denotes the trace norm. When this lower bound holds, we construct a commutator representation directly. Otherwise, the vector-selection theorem of Marcus, Spielman, and Srivastava yields smaller trace-zero compressions whose norms are small enough for the induction to close. We also construct an explicit family of zero-diagonal Hermitian unitaries that forces a lower bound of order $\sqrt{\log n}$ for $\lVert B\rVert\lVert C\rVert$ when either factor is required to be diagonal in the prescribed basis. The same family admits $\varepsilon$-pavings with fewer than $2\varepsilon^{-2}$ blocks and representations by two normal factors with optimal norm product $1/2$. This establishes a distinction between unrestricted commutator bounds and bounds under a prescribed diagonal restriction. The main results and their essential inputs are formalized in Lean 4 using Mathlib. The development also includes a formal derivation of the Kadison-Singer state-extension theorem from the same vector-selection theorem.

The Radon–Nikodym theorem for completely positive maps identifies the order interval below a map with positive contractions in the commutant of its minimal Stinespring representation. In this paper, we study the behavior of projection-valued Radon–Nikodym derivatives (sharp submaps) under composition. We prove that sharpness loss is determined by the multiplicative defect of the induced pullback map. In the finite-dimensional case, we show that this defect is determined by the orthogonal complement of the composite Kraus relation space, denoted $\mathcal{E}_{\Lambda,\Omega}$. A sharp submap remains sharp if and only if $\mathcal{E}_{\Lambda,\Omega}$ reduces the lifted Radon–Nikodym projection. Furthermore, we establish a tensor-factor criterion for the global preservation of sharpness and explicitly quantify the sharpness-loss defect.

We prove uniform approximation theorems for Chebyshev expansions of continuous functions of two variables. Under a variation-type condition on the square $[-1,1]^2$, a continuous function admits a uniformly convergent Chebyshev expansion in the first variable whose coefficients are continuous functions of the second variable. These results are applied to the spectral graph wavelet transform of time-varying signals on finite weighted graphs: the scaling and wavelet kernels are approximated by expansions with time-varying coefficients whose degrees do not depend on time, the composition of the approximate transform with its adjoint admits the same explicit coefficient formulas as in the time-independent case, and reconstruction by the pseudoinverse is stable, with explicit bounds in terms of the uniform kernel errors. Numerical experiments on a sensor network confirm the convergence and stability estimates. In a denoising problem, soft thresholding of the graph wavelet coefficients with a time-varying transform parameter improves over its time-independent, fixed-parameter counterpart.

In this paper, we construct an $n$-power quasinormal operator $T$ such that $T^n$ is not quasinormal for some positive integer $n$, thereby providing a counterexample to \cite[Lemma 3.1]{ko-filomat-2023}. We then investigate the relationships among the $n$-power quasinormality of $T$, the normality of $T^n$, and the quasinormality of $T^n$, and show that these three conditions are equivalent in finite-dimensional spaces. We also provide a new proof that $n$-power quasinormal operators have the single-valued extension property \cite[Theorem 3.2]{ko-filomat-2023}; unlike the original proof, our argument does not rely on \cite[Lemma 3.1]{ko-filomat-2023} and thus closes the gap in the original argument. In addition, for a fixed operator $T$, we characterize all positive integers $n$ for which $T$ is $n$-power quasinormal. As consequences, several results of Sid Ahmed \cite{ahmed-bmaa-2011} are extended. Finally, we prove that every paranormal $n$-power quasinormal operator is quasinormal. Closely related to this, we also give an affirmative answer to the root problem of Stanković and Kubrusly \cite[Question 2.11]{stankovic-afa-2025}.

We give a counterexample to a weak majorization problem, proposed by Z. Huang (Linear Algebra Appl., 434 (2) (2011) 457–462), for singular values of Hadamard products of nonnegative matrices. The problem asks whether, for all nonnegative matrices $A$ and $B$, \begin{equation*} \bigl\{s_j^2(A\Had B)\bigr\} \prec_w \bigl\{s_j(A\Had A)s_j(B\Had B)\bigr\} \end{equation*} holds, where $\Had$ denotes the Hadamard product, $\prec_w$ means weak majorization, and $s_j(\cdot)$ is the $j$th largest singular value of a matrix. To answer this problem, we construct two $3\times3$ entrywise positive symmetric matrices. We give rigorous upper and lower bounds for the relevant singular-value sums by Sturm's theorem, and show that the inequality fails for the partial sum with $k=2$.

Let \(q=p^h\) be an odd prime power, and let the affine group \(\mathbb F_q\rtimes\mathbb F_q^\times\) act through its canonical \((q-1)\)-dimensional irreducible representation. Qualitative matrix recovery for these rank-one orbits is known. We determine sharp lower singular-value bounds for explicit real generating windows. First, we compute the exact least singular value for every nonnegative two-level window over \(\mathbb F_{p^h}\) with \(h\geq2\), and identify the unique optimizer when \(q-1\geq10\). The resulting conditioning stays bounded away from zero on every fixed odd-characteristic tower and is within an explicit characteristic-dependent factor of the best possible value over all real windows. For every \(q=3^h\), \(h\geq2\), we construct a real three-level absolute-trace window (constant on the fibers of \(\operatorname{Tr}_{\mathbb F_q/\mathbb F_3}\)) whose least singular value is \[ \frac{q(\sqrt2-1)}{q(2-\sqrt2)-1}>\frac1{\sqrt2}. \] For the full class of real windows constant on the three trace classes, we reduce the least singular value to four scalar expressions and a symmetric \(2\times2\) matrix. This yields the exact global optimum and all equality cases: the displayed trace window is uniquely optimal up to global sign and interchange of the two nonzero trace classes. Thus zero trace mean follows from optimality. The proof combines an explicit sum-of-squares identity, uniform quadratic-form certificates, and a finite-geometric block decomposition of the orbit measurement operator.

We construct explicit algebraic finite Gabor windows that are simultaneously full spark and phase retrievable, with quantitative control of the self-ambiguity function. In every cyclic dimension, a constant–Chu window has an exactly computable ambiguity minimum of order $N^{-3/2}$; for $N\ge25$, a two-site Chu repair has margin comparable to $dN^{-3/2}$ for every divisor $d\le\sqrt N$. A quantitative algebraic regularization, based on the high-degree specialization principle used in explicit full-spark Gabor constructions, imposes full spark while retaining a fixed proportion of a seed's ambiguity margin. These ingredients give explicit simultaneous constructions in every cyclic dimension and a factor-sensitive improvement when $N$ has a divisor near $\sqrt N$. Further consequences include exact Chinese-remainder tensorization, optimal-order squarefree families, near-SIC prime regularization, lifted stability, and a finite cyclic Schrödinger construction.

This is the second in a series of papers on Connes' integration in weak Lorentz ideals. Building on the Dixmier trace theory, Birman–Solomyak perturbation theory, and strong measurability developed in Part 1, we establish a spectral form of Pietsch's correspondence for traces on these ideals, extending to this setting results of Semenov–Sukochev–Usachev–Zanin for the weak trace-class. The correspondence describes the positive normalized traces in terms of Banach limits and yields a complete spectral characterization of strong measurability. We also introduce hypermeasurability (measurability with respect to every normalized trace). Unlike the ambient ideal, it depends on the specific regularly varying function chosen, and we characterize it spectrally by means of eigenvalue sums. We further show that hypermeasurability and spectral measurability are incomparable. Finally, we apply these results to examples arising from nonclassical Weyl laws in the sense of Simon, including the logarithm of the Laplacian on a closed manifold, the double Laplacian, multi-tensor products of Laplacians, and an example arising from Connes' approach to the Riemann Hypothesis.

Let $A$ be a bounded operator on a separable right quaternionic Hilbert space and let $\CA$ be the set of compact operators commuting with $A$. We establish the formula \[ \sigb(A)=\bigcap_{K\in\CA} \sS(A+K), \] a quaternionic analogue of the classical characterization of the complex Browder spectrum. The proof rests on a factorization lemma (Lemma~\ref{lem:fact}), a consequence of the Riesz–Schauder theorem and an ascent/descent argument, and on the characterization $\sigb(A)=\sS(A)\setminus\sd(A)$ in terms of S-eigenvalues of finite type, due to Arzini and Jaatit \cite{AJ26}, for which we give here an independent proof (Lemma~\ref{lem:A}). We also record two consequences of independent interest: the intersection defining $\sigb(A)$ may be restricted, without loss, to finite-rank operators commuting with $A$ (Corollary~\ref{cor:finite}); and, by a separate argument resting on the ascent/descent stability results of \cite{KD24}, the stronger pointwise statement $\sigb(A+K)=\sigb(A)$ holds for a fixed $K\in\CA$ and arbitrary $A\in\B(\V)$ (Theorem~\ref{thm:pointwise}), known in the complex case.