ALT We study noncommutative polynomials and functions invariant under linear actions of infinite (compact Lie) groups, with a focus on SU(2) acting on two noncommuting variables. For SU(2), we give a complete description of the invariant noncommutative polynomials: there are no invariants of odd degree, and in degree 2n the invariants are exactly the span of the S₂ₙ-orbit of (XY-YX)ⁿ. We also construct an explicit basis recursively from the operation (A,B)↦ XAYB-YAXB, and show that the space of degree-2n invariants has dimension equal to the n-th Catalan number. All proofs proceed by direct computation with the Haar integral.

ALT Let ℬα denote the shift operator on the weighted Bergman space A²α(D). We prove that, for every integer k≥1, the Beurling-type theorem holds for ℬαᵏ on A²α(D) if and only if -1<α≤1. We also show that, for every α>-1, there exists a finite Blaschke product B such that M_B admits a cyclic invariant subspace but fails the wandering subspace property.

ALT The aim of this paper is to obtain the characterization of Fourier transform for functions taking values in a separable Hilbert space H, defined on the groups G=ℝⁿ, G=ℤ/nℤ. In the first part of the paper, we consider the Fourier transform as a bijective mapping on S_H, the space of all H-valued smooth rapidly decaying functions on ℝⁿ, which naturally extends bijectively to the dual space S_H'. Within this framework, we characterize the Fourier transform in terms of the convolution theorem, supplemented by additional structural properties of the Fourier transform. In the second part of the paper, we consider the discrete setting, where the Fourier transform is an operator on L¹(ℤ/nℤ, H), and we provide corresponding characterization based on its essential properties. In contrast to the classical setting, a key assumption in our framework involves the interplay between two distinct convolution operations. As a consequence, the structure of our proof diverges notably from the classical ca

ALT We characterize when a de Branges–Rovnyak space coincides with a harmonically weighted Dirichlet space. We obtain a necessary and sufficient condition for this identification, which settles a long-standing open problem. We illustrate the result with several examples, that include measures supported on uncountable sets.

ALT Let R∞(n) be the maximal Banach–Mazur distance from an n-dimensional real normed space to ℓ∞ⁿ. We prove that R∞(n)≥ cn²/³(log(2n))⁻¹⁰. Together with the upper bound of Friedland, Naor, and Youssef, this gives R∞(n) = n²/³⁺ᵒ⁽¹). The examples are polars of symmetric Gaussian polytopes with 2n generators, and the lower bound holds with exponentially high probability. A joint minimization over coefficient representations and positive weights yields an exact second-variation identity and sparse coefficients. A Gaussian projection argument controls all low-rank completions of the recorded matrix. An adaptive choice of exceptional generators then reduces the remaining case to a fixed-shift Gaussian small-ball estimate over two finite nets.

ALT The complex polynomial Bohnenblust–Hille inequality bounds the coefficient ℓ₂ₘ/(ₘ₊₁) norm of a complex m-homogeneous polynomial by its supremum norm, with a constant independent of the number of variables. For each fixed integer N≥3, we characterize the choices of N degree-m monomials for which the inequality holds with optimal constant 1 for every polynomial supported on them, provided m exceeds an explicit threshold depending only on N. This holds precisely when their exponent vectors contain no nontrivial three-term arithmetic progression; equality for a nonzero polynomial then holds only for monomials. We also prove that, in the same degree range, every polynomial with at most N nonzero coefficients that violates the constant-one inequality has a unique coefficient of maximal modulus, whose exponent is the center of such a progression in its support. Exact contractivity also follows from a comparability condition on the two largest coefficients, with a degree threshold that necessa

ALT We construct a bounded closed convex set in c₀(ℂ) on which no nonzero continuous complex-linear functional attains its maximum modulus. Consequently, this answers in the negative the two longstanding questions raised by Deville, Godefroy and Zizler inp. 35]DGZ. The construction uses a small perturbation of the identity on c₀(ℝ).

ALT Let 1≤ p<∞. We prove an Lᵖ-operator analogue of Raeburn's symmetric imprimitivity theorem for countable discrete groups. Let A be an Lᵖ-operator algebra with a contractive two-sided approximate identity, admitting a nondegenerate isometric representation on a σ-finite Lᵖ-space. Given commuting free and proper actions of groups G and H on a second-countable locally compact Hausdorff space Z, together with commuting isometric actions on A, we show that the two corresponding full crossed products, and likewise the two reduced crossed products, are Morita equivalent as Banach algebras. The proof is organized through the linking groupoid. Its main analytic ingredient is a coefficient-valued spatial matrix dilation; in the reduced case we prove that this dilation preserves regularity after a Borel reindexing of the opposite corner. A pairing-valued approximate identity yields fullness and nondegeneracy.

ALT Noninjective multiplication symbols allow commuting operators to mix function values along their fibers, while Hölder regularity constrains this mixing at branch points. We study these operators on the Banach algebra Aᵦ, 0<β<1, defined by a restricted interval-variation condition. For u(t)=t(1-t), we determine the full commutant {Mᵤ}'=mathscr Cᵦ, including operators whose reflection coefficients need not belong to Aᵦ. Separately, we characterize the operator-norm closure Jᵦ of the regular multiplication–reflection subalgebra of mathscr Cᵦ, obtain two-sided norm-distance estimates, and find a complemented copy of ℓ∞/c₀ in the Banach-space quotient mathscr Cᵦ/Jᵦ. Every T∈mathscr Cᵦ satisfies σ(T)=σₐₚ(T)=σ_usf(T)=σₑₛₛ(T), and Fredholmness is equivalent to invertibility. For multipliers with finitely many full bi-Lipschitz branches, we also characterize their full commutants and establish the same spectral equalities for every commuting operator. For the regular finite-branch algebra, we e

ALT We prove that a Lipschitz-free space \F(M) fails Kwapień's property (K) whenever its underlying metric space M contains a bi-Lipschitz copy of C× K, where C⊂ℝ is closed with positive measure and K contains a Cantor metric space. Consequently, this holds for every subset of ℝⁿ, n≥slant2, whose closure has positive measure, and for a compact planar set of Assouad dimension one. These Lipschitz-free spaces embed into no strongly weakly compactly generated space and no L₁-space; in the separable case their duals are not isomorphic to ℓ∞. Combining our result with a decomposition of Aliaga and Medina, we show that \F(ℝⁿ×ℤᵐ) is strongly weakly compactly generated if and only if it has (K), and this occurs precisely when n≤1. In particular, \F(ℝ²) does not embed isomorphically into \F(ℝ×ℤ), answering a question of Aliaga. The planar examples nevertheless have the Dunford–Pettis property and Pełczyński's property (V*) and are weakly sequentially complete.

ALT Given a minimal system E for a Banach space X, Casazza, Dilworth, Kutzarova and Motakis introduced a multiplication ⊗ on span(E) and studied the Schauder bases for which ‖x ⊗ y‖≈ ‖x‖‖y‖ asking whether every such basis is equivalent to the canonical unit vector basis of c₀ or ℓₚ. We continue this study in the more general setting of minimal systems that are not assumed to be Schauder bases, and we obtain positive answers under hypotheses that are formally weaker than the ones considered so far: partial democracy instead of democracy or bidemocracy, and truncation quasi-greediness instead of quasi-greediness. Along the way we show that the above multiplicative estimate forces invariance under translations, we characterize when such a minimal system is a Schauder basis, and we give examples showing that the one-sided variants of the estimate behave very differently.

ALT For a function ψ: mathbbR^d → mathbbC, we define the phase-corrected gradient ∇_circ ψ=|ψ|^-1 overlineψ ∇ψ and study the regularity of the map ψmapsto ∇_circ ψ on Sobolev spaces. Since Re[∇_circ ψ] = ∇ |ψ|, the phase-corrected gradient fits naturally into the framework of nonlinear composition operators. Its imaginary part Λ=Im[∇_circ ψ], on the other hand, plays a key role in quantum hydrodynamic models. We prove that ψmapsto ∇_circ ψ is bounded and continuous from W^s,2(mathbbR^d) to W^s-1,2(mathbbR^d) for 1 ≤ s < ³⁄₂, and we show that the threshold s = ³⁄₂ is sharp. In particular, this answers an open question concerning the continuity of the map ψmapsto |ψ| on Sobolev spaces. We also show that our results remain valid for functions that do not necessarily decay at infinity, a relevant setting in the context of Ginzburg–Landau theory. Finally, we extend our analysis to more general phase-corrected gradients of the form f(|ψ|^-1 ψ) ∇ ψ, where f: S¹ → mathbbC is Lipschitz, and discuss

ALT We study the operator-norm approximation of Markov operators associated with finite graph coverings. Let q:Y→ X be a connected covering of degree d≥ 2 between finite connected graphs, and let Aq:C(Y)→ C(X) be its fiber-averaging operator. We show that Aq is not a finite convex combination of extreme Markov operators, although such combinations approximate Aq arbitrarily well in operator norm. We determine the exact distances from Aq to convex combinations of at most m extreme Markov operators, with arbitrary or equal convex weights. If m=kd+r, where k≥ 0 and 1≤ r≤ d, these distances are 2(d-r+1)/[d(k(d-r+1)+1)] and 2r(d-r+1)/(dm), respectively. Both distances are attained and depend only on the covering degree d and the number of terms m. We also determine when equal weights are optimal and quantify the additional approximation error when they are not. Our proofs use total-variation estimates and continuous rearrangements of labelled weighted atoms. In particular, the unrestricted erro

ALT Let AΨω(D) be the weighted Bergman-Orlicz spaces induced by doubling weights ω on the unit disk D. Suppose Orlicz function Ψ satisfies the Δ₂-condition. In this paper, the compactness of finite linear combination of composition operators on AΨω(D) are completely characterized in terms of Julia-Carathéodory type conditions and vanishing Carleson measure, respectively.

ALT The symplectic interlacing theorem states that the symplectic spectrum of a 2n × 2n real symmetric positive definite matrix satisfies a 2k-step interlacing relation with the symplectic spectrum of every 2(n-k) × 2(n-k) s-principal submatrix. We prove the converse in this paper. More precisely, given a positive global symplectic spectrum of size n and a positive local symplectic spectrum of size n-k satisfying the 2k-step interlacing inequalities, we construct a 2n × 2n real symmetric positive definite matrix having the prescribed global symplectic spectrum and a 2(n-k) × 2(n-k) s-principal submatrix of it having the prescribed local symplectic spectrum. Our main tool is a reduction of s-principal compression to a rank-k singular value perturbation. In the context of the Gaussian quantum marginal problem, our result implies that the symplectic interlacing inequalities are not merely necessary conditions on Gaussian marginals, but are also the complete spectral compatibility conditions f

ALT We study how a functional inequality satisfied by the one-particle conditional distributions of a Gibbs measure, with a constant independent of the positions of the other particles, can be propagated to the whole N -particle system, with a constant independent of N, when the interaction is singular. Our model is the Coulomb gas on the torus T 3 in the mean-field regime. We distinguish two mechanisms. The first one is a dependent tensorization, governed by Dobrushin-Wu interdependence coefficients. It yields uniform Poincaré and transport-information W 2 I inequalities with fully explicit constants. We explain why this mechanism does not extend to the log-Sobolev inequality: the Zegarlinski-type coefficients, which require pointwise bounds on the interaction, are infinite. The second mechanism is a tensorization with a defect. Using a uniform bound on the partition function due to Duerinckx and Jabin, it yields a uniform log-Sobolev inequality for the Coulomb gas on T 3 at small charge.

ALT Alshehri and Lykova (J. Geom. Anal. 33 (2023)) formulated a Schwarz lemma for the pentablock P, giving a criterion for the existence of an analytic map from the unit disc D into the closed pentablock P̅ that sends the origin to the origin and a prescribed point λ₀∈D to a given point x₀∈P̅. We show that the sufficiency direction of their criterion is false. We identify the underlying error as the extension to the open unit disc of a Fejér–Riesz identity that is valid only on the unit circle. We establish a corrected necessary and sufficient interpolation criterion when the last two components are prescribed by a fixed rational Γ-inner function and resolve the special case s₀=0. Using the Poisson integral representation of the associated outer spectral factor, we obtain a sharp comparison between the corrected bound and the bound proposed by Alshehri and Lykova, with a precise characterization of equality. We further express the logarithmic difference between these bounds as a nonnegativ

ALT We introduce the concept of a fractional upper gradient and employ it to characterize the p-Muckenhoupt weighted s-fractional p-Sobolev spaces on ℝᵈ where d∈ℕ, 1<p<∞ and 0<s<1.

ALT We determine the Serre closure of Banach spaces in the category of quasi-Banach spaces, that is, the smallest class of quasi-Banach spaces that contains all Banach spaces and is stable under subspaces, quotients, and extensions. Precisely, a quasi-Banach space X belongs to this closure if and only if there exist constants M and k∈mathbbN such that ‖x₁+…+x_n‖/‖x₁‖+…+‖x_n‖≤ Mlog^k n for every n≥ 2 and every x₁,…,x_n∈ X; equivalently, X is a quotient of an Orlicz sequence space \ell(I,f_l), where I is an index set and f_l(t) is equivalent to tlog^l(1/t) near zero. The proof combines geometrical estimates for quasi-Banach spaces and quasilinear maps with the study of higher order derived spaces associated to complex interpolation between \ell₂/3 and \ell₂. We provide examples of spaces in the Serre closure of Banach spaces that are not iterated extensions of Banach spaces and discuss some related problems.

ALT In this paper we introduce a class of operators which we call DW-L-weakly compact operators. They carry disjointly weakly compact subsets of a Banach lattice onto L-weakly compact sets. The class of DW-L-weakly compact operators is closely related to two recently introduced classes of operators: almost L-weakly compact operators and order L-weakly compact operators. We show that DW-L-weakly compact operators are precisely those operators which are both almost L-weakly compact and order L-weakly compact. The adjoint and the order structure of DW-L-weakly compact operators are also considered.