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Wednesday, 4 November 2026: 12 new papers, 3 cross-lists and 13 replacements in Functional Analysis (math.FA), in arXiv's order, with full abstracts. The same papers as arXiv's own listing, set as a newspaper; every title links to arXiv.
We extend Brown's one-variable rigidity theorem to strictly convex functions of two variables under a regularity assumption. Let $K\subset\mathbb{R}^2$ be nonempty, compact, and convex, and let $f$ be real-valued, strictly convex on $K$, and of class $C^2$ on a neighbourhood of $K$. We prove that the generating set $\{x_1,x_2,f\}$ is hyperrigid in $C(K)$.
We characterize bounded small Hankel operators from $B_p(\om)$ to $\overline{B_q(\om)}$ on the polydisc for $0<p,q<\infty$, under explicit conditions relating the kernel parameter to the weight. When $q<p$, boundedness is equivalent to a weighted Besov condition on the symbol; when $p\le q$, it is equivalent to a weighted supremum condition on a shifted radial derivative. In particular, the diagonal case is characterized by the product Bloch space. The proof combines weighted Bergman duality and atomic factorization with local kernel estimates and randomized atomic tests.
By a theorem of Bagchi, the Riemann Hypothesis (RH) holds if and only if $E(s) = 1/s$ belongs to the closed linear span of $G_k(s) = (k^{-s} - k^{-1})\zeta(s)/s$, $k \geq 2$, in the Hardy space $H^2(\mathbb{C}_{1/2})$, where $\mathbb{C}_\alpha$ is the half-plane $\mathrm{Re}(s) > \alpha$. The formal M\"obius expansion $E = \sum_{k \geq 2} \mu(k) G_k$ diverges on the critical line, by a theorem of B\'aez-Duarte. We study it in the scale of spaces $H^2(\mathbb{C}_\alpha)$, where its convergence forces $\zeta$ to be zero-free in $\mathbb{C}_\alpha$. We prove that the partial sums $S_n$ converge to $E$ in $H^2(\mathbb{C}_\alpha)$ for every $\alpha > 1$, and that a single zero of $\zeta$ in $\mathbb{C}_\alpha$ keeps the best-approximation error $d_n(\alpha)^2$ uniformly away from zero. We express the M\"obius error $s_n(\alpha) = \|S_n - E\|^2_\alpha$ as a series with elementary terms, free of $\zeta$, and compute it for $n \leq 10^6$: $s_n(1/2)$ oscillates without decaying, whereas for $\alpha > 1/2$ it decays at a rate close to $n^{1-2\alpha}$, much faster than the bound known under RH. On the critical line, $d_n(1/2)^2$ behaves like $0.046/\log n$ for $100 \leq n \leq 5120$, in agreement with earlier computations by B\'aez-Duarte, Balazard, Landreau and Saias.
Let $H$ be a complex Hilbert space. We prove that the class of pseudo-Fredholm operators on $H$ is a regularity in the sense of Kordula and M\"uller. We also prove that the classes of left and right generalized Drazin invertible operators are regularities on $H$. Consequently, the generalized Kato spectrum and the corresponding generalized Drazin spectra satisfy the spectral mapping theorem. We construct an operator $T$ on a separable complex Banach space such that $T^2$ is right generalized Drazin invertible whereas $T$ is not pseudo-Fredholm. In particular, the power axiom need not hold for pseudo-Fredholm operators on Banach spaces. Finally, we show that the accumulation sets of the approximate point and surjectivity spectra define regularities on Banach spaces, yielding another proof of the regularity of generalized Drazin invertible operators in $\B(X)$.
The upper bild of the quaternionic numerical range provides a convex representative of the generally nonconvex quaternionic numerical range. Although its convexity is known, the geometry of the upper bild is difficult to describe explicitly, even for 2-by-2 quaternionic matrices. In this paper, we revisit the upper bild of quaternionic numerical ranges from both algebraic and geometric viewpoints. We first point out gaps in Kumar's proof. We then investigate several special classes of 2-by-2 quaternionic matrices and describe their shapes and boundary structures in the closed upper complex half-plane directly in terms of the standard eigenvalues and the off-diagonal entry of the Schur form. For the general 2-by-2 case, we use a different canonical form. Following So and Thompson, we prove the concavity of the upper-boundary function through the associated complex representation. For the lower boundary, we obtain a distance-to-a-disk representation and use convex optimization to prove its convexity. We also treat the degenerate cases. Together with the connectedness of the vertical sections, these results give an alternative proof of the convexity of the upper bild for arbitrary quaternionic matrices.
We study subspaces of two-Lipschitz mappings using group actions and tensor preduals. We first show that the ambient spaces contain large classical sequence spaces. We then use similarity actions to select smaller subspaces. Character-equivariant mappings form weak$^*$-closed fixed-point spaces and admit quotient tensor preduals. We give two applications. For mappings on Banach spaces, we describe positive homogeneity in each variable and construct bounded projections. For mappings on finitely generated groups, we characterize harmonicity in each variable. We show that the harmonic predual is the projective tensor product of the corresponding one-variable quotient preduals.
We study $\mathcal{F}$-hypercyclicity of weighted orbits $\lambda_nT^nx$ for continuous linear operators $T$ on complex F-spaces, where $\mathcal{F}$ is a Furstenberg family with the cut--shift--and--paste (CuSP) property. We characterize the scalar sequences for which weighted hypercyclicity always implies ordinary hypercyclicity: their nonzero values must be bounded and bounded away from zero. The same condition removes weights for every such family. For the converse inclusion, we prove that a bounded scalar sequence preserves all $\mathcal{F}$-hypercyclic vectors if its backward-shift $\omega$-limit system is distal and does not contain the zero sequence. For chaotic operators, minimality of this system suffices. In particular, every complex nilsequence which is not identically zero preserves $\mathcal{F}$-hypercyclicity, and equality holds when its nonzero values are bounded away from zero. We also obtain simultaneous weighted return results and stability under unimodular weights with asymptotically trivial multiplicative differences.
The aim of this paper is to characterize the Fourier transform of functions taking values in a separable Hilbert space $\Hi$ and defined on locally compact abelian groups. Inspired by Krukowskis characterization of integral transforms on locally compact abelian groups, we establish characterizations of the Fourier, Laplace, and cosine transforms for separable Hilbert space-valued functions on locally compact abelian groups. Furthermore, we obtain a characterization of the group Fourier transform on the Heisenberg group.
We use real-analytic functional calculus to show that the envelope of the algebra of $m$-variable polynomials with real coefficients, with respect to the class of strictly real Banach algebras, is isomorphic to the algebra of real-valued analytic functions on $\mathbb{R}^m$.
Let $\alpha\in\mathbb R$, $0<p<\infty$, $0<q,r\le\infty$, and $W$ be a matrix $\mathcal A_p$ weight on ${\mathbb R}^n$. We introduce the matrix-weighted $Z$-spaces $Z^{p,q,r}_\alpha(W)$ and tent spaces $T^{p,q,r}_\alpha(W)$ on the upper half-space and establish their characterizations using reducing operators. We then characterize homogeneous matrix-weighted Besov and Triebel--Lizorkin spaces in terms of the corresponding $Z$-spaces and tent spaces, respectively, using Schwartz kernels satisfying Tauberian and finite moment conditions. Furthermore, we establish duality for the matrix-weighted $Z$-spaces and tent spaces and obtain real and complex interpolation results for these spaces and the associated homogeneous matrix-weighted Besov and Triebel--Lizorkin spaces.
We investigate explicit and implicit dissipative Hamiltonian representations of generators of strongly continuous semigroups on real or complex Hilbert spaces. For a prescribed positive definite selfadjoint energy operator $H$ satisfying $\operatorname{dom}(A)\subset\operatorname{dom}(H)$, we show that the Lyapunov dissipation inequality is equivalent to an exact factorization $A=DH$, where $D$ can be chosen maximally dissipative. For bounded $H$, the criterion also yields an implicit representation. We further study energy constructions based on closed intertwiners and output trajectories, distinguishing contractive energy-space dynamics from operator factorizations on the original state space. Examples illustrate the roles of noncoercivity, unboundedness, and domain compatibility.
We prove two sufficient conditions for the frame property of Gabor systems $\mathcal{G}(g,a,b)=\{e^{2\pi i mb\,\cdot}g(\cdot-ak):k,m\in\mathbb{Z} \}$ generated by polynomial--Gaussian windows $g(t)=P(t)e^{-ct^2}$, where $c>0$ and $P$ is a nonzero complex polynomial of degree $n$. The conditions depend only on the product $ab$ of the lattice parameters and the degree of $P$. First, $\mathcal{G}(g,a,b)$ is a frame whenever $ab<1$ is irrational. Second, if $ab=p/q<1$, with $p,q\in\mathbb{N}$ coprime, then $\mathcal{G}(g,a,b)$ is a frame whenever $q-p>n$. Consequently, in the region $ab<1$, failure of the frame property can occur only on rational hyperbolas whose reduced denominator and numerator differ by at most $n$. The proof extends the translated-Wronskian method of Faulhuber and Petersen [\emph{The frame set of the first Hermite function}, arXiv:2609.02610] from the first Hermite function to arbitrary polynomial multiples of a Gaussian, combining a multiplicity estimate for Wronskians with Jensen's formula.
We study the self-consistent common-noise equation associated with the Weil--Petersson response differential of the rotationally marked SLE welding probability. For $0<\kappa\leq4$, we prove joint weak uniqueness among continuous adapted weak solutions whose law at each deterministic time is the welding probability, and obtain the Markov property in the original filtration. A conditional Gaussian lift and capacity localization give the required $L^1$ generator-range density. Identification with the canonical law gives stationarity and reversibility. We also construct autonomous Fej\'er and Poisson regularizations with pathwise-unique global strong solutions. Their joint state-noise laws converge to the canonical law for every initial probability absolutely continuous with respect to the welding probability.
We prove a space-time analogue of the HWI inequality of Otto-Villani for controlled kinetic equations with a semiconvex confining potential. It bounds the time-averaged relative entropy by spatial and conditional velocity transport distances, the velocity Fisher information, an intrinsic action of the path, and a boundary term given by the time endpoint values of the Wasserstein current. The key step is a differential comparison inequality for the Wasserstein current, which captures the transfer of dissipation from velocity to position. Along the Langevin semigroup, this inequality yields a kinetic evolution variational inequality involving the second time derivative of the squared spatial Wasserstein distance and a damping term. Under a compatible spatial Talagrand's transport inequality, we obtain a boundary-free space-time logarithmic Sobolev inequality on every interval $[0,T]$, $T>0$, with the optimal $T^{-2}$ short-time scaling of the constant. As a consequence, we also establish entropy decay for every friction coefficient and every initial density of finite entropy, with optimal cubic order at short times and exponential decay at long times.
A space is ESLC if it contains neither an $S$-space nor an $L$-space, and NFS is the class of spaces containing no free sequences of length $\omega_1$. We prove that under $\mathrm{MA}(\omega_1)$, if all finite powers of $X$ belong to NFS, then $C_p(X)^\omega$ is ESLC; thus, for subspaces of $C_p(X)$, hereditary Lindel\"ofness, hereditary separability and countable spread coincide. This is derived from a dual form of a theorem of Todor\v{c}evi\'c. A Lindel\"of $\Sigma$-space belongs to NFS if and only if its compact subspaces have countable tightness. Hence $C_p(X)$ contains no $L$-spaces if $X$ is a Lindel\"of $\Sigma$-space of countable tightness and $\mathrm{MA}(\omega_1)$ holds, or if $X$ is any Lindel\"of $\Sigma$-space and PFA holds. Under $\mathrm{MA}(\omega_1)$, all iterated function spaces of Corson compact spaces are ESLC, and in ZFC the space $C_p(X,\{0,1\})$ is ESLC for every zero-dimensional Corson compact space $X$.