Papers

We introduce several notions of resonance for systems exhibiting weak or partial dissipation: $\dot u=Au+f(t)$, with $A$ the generator of a strongly stable semigroup $S(t)$ on a Hilbert space $H$. We refer to resonance as the phenomenon where a time-periodic $f$ may yield an unbounded $u$, classically occurring when $A$ has imaginary eigenvalues. In infinite dimensions, however, unbounded growth may depend on topological choices, and several “resonances" may occur. Classically, existence of $T$-periodic $u$ under $T$-periodic $f$ is equivalent to a range condition: $\mathcal{R}(I-S(T))=H$. Consequently, its failure permits unbounded solution growth and we elucidate that connection through properties of $S(t)$. We describe a hierarchy of range failures, and provide an associated resonance taxonomy. The growth rate of ~$||(A-\lambda I)^{-1}||_H$ on ~$i\R$ dictates a regularity gap between $u$ and $f$, and, for rapid growth in $\lambda$, that gap can be infinite. In this case, we demonstrate implications for periodic solvability, and a mechanism for constructing smooth resonant forces. The theory developed here is motivated by (and demonstrated for) a hyperbolic-parabolic system, where the latter component provides the only system dissipation. Such dynamics are a simplification of fluid-structure phenomena, which demonstrate strong stability but do not provide unconditional periodic well-posedness. Though point-spectral resonance is ruled out, we note that $\mathcal R(I-S(T))\neq H$. Our main result here shows that, for a carefully constructed geometry, a classical heat-wave system possesses periods $T$ which yield infinite derivative loss and, via supporting results, yield resonance with smooth forcing.

Barycentric Rényi divergences were introduced in [Mosonyi, Bunth, Vrana, Linear Algebra and its Applications, 2024] as an alternative to standard Kubo-Ando constructions to define multivariate quantum Rényi divergences. They are defined via a variational expression and depend on a finite collection of quantum relative entropies $D^{q_x}$. When all the relative entropies are monotone under CPTP maps then so are the corresponding barycentric Rényi divergences, and when all the relative entropies are additive then the corresponding barycentric Rényi divergences are subadditive under tensor product. Additivity has only been established before for the case where all $D^{q_x}$ are chosen to be the Umegaki relative entropy, which is also the only case where the barycentric Rényi divergence (called the minimal one) was known to admit an explicit expression. Here we settle the problem of additivity by showing that for any choice of monotone quantum relative entropies, the regularized barycentric Rényi divergence coincides with the minimal barycentric Rényi divergence on invertible inputs. This in turn implies that if at least two of the $D^{q_x}$ are strictly larger than the Umegaki relative entropy (in a precise sense), then the resulting barycentric Rényi divergence is not even weakly additive.

We study the singular limit $p \downarrow 1$ for positive radial entire solutions $u_p$ of $$ -\Delta_p u_p=F_p\left(|x|, u_p,\left|\nabla u_p\right|\right) u_p^{-\beta_p} \quad \text { in } \mathbb{R}^n, $$ where $n \geq 2,1<p<\min \{2, \sqrt{n}\}$, and $0 \leq \beta_p \leq p-1$. Assuming that $$ F_p \rightarrow F_1 \quad \text { locally uniformly }, \quad \frac{\beta_p}{p-1} \rightarrow \mathbf{c} \in[0,1] \quad \text { as } p \downarrow 1, $$ we prove that, after passing to a subsequence, $u_{p_j} \rightarrow u$ in $C_{\text {loc }}\left(\mathbb{R}^n\right)$ and $\nabla u_{p_j} \stackrel{*}{\rightharpoonup} \nabla u$ in $L_{\text {loc }}^{\infty}\left(\mathbb{R}^n ; \mathbb{R}^n\right)$, where $u \in W^{1, \infty}\left(\mathbb{R}^n\right)$ is radial, and solves $$ -\Delta_1 u=\mathtt{K}(|x|) \quad \text { in } \mathbb{R}^n . $$ On $\left\{x \in \mathbb{R}^n \mid \mathtt{G}(|x|)<1\right\}$, we prove that $\mathtt{K}(|x|)=w^{-\mathbf{c}} F_1(|x|, u(x), 0)$ and that $\nabla u=0$ a.e., where $u_{p_i}^{p_i-1} \rightarrow w$. On the other hand, we show that if $\rho \mapsto F_1(r, t, \rho)$ is affine for every $r$ and $t$, then $\mathtt{K}(|x|)=$ $w^{-\mathbf{c}} F_1(|x|, u(x),|\nabla u(x)|)$ a.e. in $\mathbb{R}^n$. For $n=1$ and $1<p<2$, we study $$ \frac{\mathrm{d}}{\mathrm{~d} x}\left(\left|u_p^{\prime}\right|^{p-2} u_p^{\prime}\right)=\mathtt{F}_p\left(|x|, u_p,\left|u_p^{\prime}\right|\right) u_p^{-\beta_p} \quad \text { in } \mathbb{R} $$ and obtain analogous results as $p \downarrow 1$.

A sequence $(y_n^*)$ in the unit sphere of $E^*$ is strongly normal if the vectors $x\in E$ satisfying $\sum_n|y_n^*(x)|<\infty$ form a dense subspace. For every set $\Gamma$ of cardinality at least the continuum, we construct a normalized weakly null sequence in $\ell_1(\Gamma)^*$ with no strongly normal subsequence. This answers a question of Śliwa negatively in ZFC. Together with the classical selection argument below the bounding number $\mathfrak b$, the construction gives positive and negative bounds for this subsequence property on $\ell_1(\kappa)$. Its validity on $\ell_1(\omega_1)$ is independent of ZFC. The intermediate range $\mathfrak b\leqslant\kappa<\mathfrak c$ remains undecided by these results. We also give a counterexample in a Banach sequence space that is not isomorphic to any $\ell_1(\Gamma)$.

We prove that pseudo-amenability implies approximate amenability for Banach algebras with a bounded approximate identity. Consequently, these two notions of generalized amenability are equivalent for such Banach algebras. Our proof closes a gap in the paper by F. Ghahramani and Y. Zhang, Pseudo-amenable and pseudo-contractible Banach algebras.

We give a variational comparison proof of the sharp logarithmic Sobolev inequality with constant $2$ for the word-length Poisson semigroup on free group von Neumann algebras. Our approach extends the Lagrange method for cyclic groups in [Yao26] to a framework in finite tracial von Neumann algebras. The inequality was recently established by Xie and Zhang in [XZ26] using a completely different approach.

We construct two complemented subspaces of $L_1[0,1]$. The first has the Schur property but fails the Radon–Nikodým property. The second contains a copy of $\ell_2$ but no copy of $L_1[0,1]$. Neither space is isomorphic to a Banach lattice. This gives a negative answer to the complemented-subspace question of Lindenstrauss and Rosenthal. A Lean 4 formalisation of the main results accompanies the paper.

We give a deterministic polynomial-time algorithm that, given rational Hermitian matrices $H_1,\dots,H_N$ of rank at most one, finds signs $s\in\{\pm1\}^N$ with $\|\sum_i s_i H_i\|\le 13\|\sum_i H_i^2\|^{1/2}$. As a corollary, for vectors $v_i$ with $\sum_i v_iv_i^*=I$ and $\|v_i\|^2\le\delta$, the signs yield a partition $[N] = S_1 \cup S_2$ such that each part satisfies $\|\sum_{i \in S_j} v_i v_i^* - \frac{I}{2}\| \leq \frac{13}{2}\sqrt\delta$ for $j = 1,2$. This gives a deterministic polynomial-time algorithm for the Kadison–Singer problem, in Weaver's equivalent discrepancy-theoretic $\mathsf{KS}_2$ formulation, with a universal constant.

Let $\mathcal{S}^ r$ denote the Schatten–von Neumann class and let $S_{\Psi_{f,\lambda}}$ be the Schur–Hadamard multiplier whose symbol is the divided-difference matrix of $f$ along $\lambda$. Let $0<\alpha,r<\infty$, let $f\in C^1([-1,1])$ satisfy $f(0)=0$, $|f'(t)|\lesssim |t|^\alpha$, and let $\lambda\in\ell^r$ be real with $\|\lambda\|_{\ell^\infty}\leq1$. We determine the pairs $0<p,q\leq\infty$ for which $S_{\Psi_{f,\lambda}}:\mathcal{S}^ q\to\mathcal{S}^ p$ is bounded for every such $f$ and $\lambda$. Our principal new positive estimates treat $q=\infty,1$ and $0<p<1$. Precisely, we show that the boundedness holds exactly when \[ \frac1p\leq\frac{\alpha}{r} +\min\!\left\{1,\frac1q\right\}. \] In particular, this resolves the untreated cases in [Arazy, PAMS, 1982] and [Potapov, Sukochev, Tomskova, Adv. Math., 2015]. Our method also delivers a new proof of the main results in just cited papers.

We develop a theory of Orlicz and Lorentz–Zygmund sequence spaces. We study in detail elementary properties of both structures. In particular, we characterize the values of governing parameters for which the corresponding spaces satisfy the axioms of rearrangement-invariant Banach sequence spaces, we evaluate the asymptotic behavior of their fundamental functions, and we characterize their associate spaces. In our main result, we characterize all cases in which a Lorentz–Zygmund sequence space coincides with some Orlicz sequence space.

This paper systematically investigates the structural, analytical, and operator-theoretic properties of the Nevanlinna class $\mathcal{N}_u$ of Dirichlet series, introduced by Brevig and Perfekt [Adv.\ Math., 2021] and further developed by Guo et al.\ [Ann.\ Inst.\ Fourier (Grenoble), 2025]. First, we examine the topological structure of $\mathcal{N}_u$. We then prove a Littlewood–Paley type identity for functions in $\mathcal{N}_u$, establish an equivalent characterization via vertical limit functions, and demonstrate that the interchange of limits in this identity is permissible. In addition, we provide an alternative proof of this identity using potential-theoretic approach. Applying these analytical tools, we study composition operators $C_\Phi$ acting on $\mathcal{N}_u$ and characterize those symbols $\Phi$ that induce bounded composition operators $C_\Phi$. Utilizing Carleson measure techniques on half-planes and infinite-dimensional tori, we characterize the symbols $\Phi$ that generate bounded and compact composition operators. In particular, we prove the complete equivalence between the (vanishing) Carleson embedding condition and the geometric (vanishing) Carleson condition on Carleson squares. We conclude with two function-theoretic applications to functions in $\mathcal{N}_u$.

Partial sums of the Taylor expansions of functions from the Neil algebra are uniformly bounded by their uniform norms on the ball of radius $r$ centered at the origin, where $r$ denotes the unique real root of \[ 4x^3 + x - 2 = 0. \] Moreover, this radius $r$ is optimal.

We continue to develop a theory of Besov and Triebel-Lizorkin spaces associated with weighted subcoercive operators on a real connected Lie group, specializing to the case of groups with polynomial volume growth. We consider the full scale of spaces and consider equivalent definitions, inclusions, discretization, and duality.

Let $\Delta_k$ be the Dunkl Laplacian associated with an arbitrary root system and a nonnegative multiplicity function. For every $0<p\leq1$, a Coifman-Weiss type atomic characterization of the Hardy space $H^p_{\mathrm{Dunkl}}$ is established. More precisely, it is proved that the space $H^p_{\mathrm{Dunkl}}$, which is originally defined by a relevant square function, coincides with the space generated by $({\rm CW},p,2)$-atoms, that is, atoms supported on Euclidean balls and satisfying cancellation conditions against all polynomials of degree $ \leq s_p$, where \[ s_p=\left\lfloor \mathbf N\left(\frac1p-1\right)\right\rfloor \] and $\mathbf N$ is the homogeneous dimension of the underlying Dunkl measure. The corresponding quasi-norms are equivalent. We also show that the same space is obtained when the $L^2$ size condition in the definition of atoms is replaced by the $L^\infty$ size condition. The strategy of the proof is to use an operator-type atomic decomposition associated with the Dunkl Laplacian, and then prove that each such operator atom can be written a linear combination of $({\rm CW},p,2)$-atoms.

Let $U,V$ be matrix $\mathcal A_p$ weights, $1<p<\infty$, and let $B$ be a matrix-valued function. For Calderón–Zygmund operators $T$ with scalar standard kernels, we characterize boundedness and compactness of $[M_B,T\otimes I_m]:L^p(U)\to L^p(V)$ by two-matrix BMO and vanishing oscillation conditions. The converse statements require non-degeneracy condition. For matrix \(\mathcal A_2\) weights \(U,V\) and a fixed Riesz transform \(R_j\) on \(\mathbb R^d\), \(d\geq2\), we characterize the Schatten class membership of \(C_{B,R_j}:L^2(U)\to L^2(V)\). We characterize $\mathcal S^r$ membership for $r>d$ and $\mathcal S^{d,\infty}$ membership by $\ell^r$ and weak $\ell^d$ conditions on the two Hilbert–Schmidt cube oscillations. For $0<r\leq d$, $\mathcal S^r$ membership is equivalent to $B$ being constant almost everywhere. The weak endpoint condition implies a weighted first-order Sobolev estimate for $C^1$ symbols, with a converse for constant or uniformly elliptic weights.

We consider the harmonically weighted Dirichlet spaces $D(\mu)$ induced by Borel measures which are mutually absolutely continuous with respect to the Lebesgue measure $m$ on the unit circle $\mathbb T$. Let $H^2$ and $D$ denote the Hardy space and the Dirichlet space on the unit disc $\mathbb D$, respectively. For any $f\in H^2$, let $m_f$ denote the measure defined by $dm_f(\zeta)=|f(\zeta)|^2 dm(\zeta)$. Our first result provides a characterization of functions in $D(m_f)$, when $f$ satisfies the condition $|f'(z)|^2=O(\frac{1}{(1-|z|)^{1-\epsilon}})$ for some $\epsilon>0$. We then study the multiplier algebras of these spaces with particular emphasis on inner multipliers. Specifically, when $f'$ is bounded, we obtain an explicit description of all singular inner functions in the multiplier algebra of $D(m_f)$ in terms of the radial zero set of $f$. As an application, we prove that if $f',g'$ are bounded and the radial zero sets of $f$ and $g$ are different, then the operators $(M_z, D(m_f))$ and $(M_z, D(m_g))$ are not similar. This generalizes a result of Richter showing that the operators $(M_z, D(m_{z-1}))$ and $(M_z, D)$ are not similar. Then, we classify all invariant subspaces $\mathcal M$ of $(M_z, D(\mu))$ for which $M_z|_{\mathcal M}$ is similar to $(M_z, D(\mu))$, where $\mu$ is any measure mutually absolutely continuous with respect to $m$. Finally, we study a special case of the problem of finding all functions $g\in D$ for which $gD(m_g)$ is a closed invariant subspace of $(M_z, D)$. A key ingredient in many of our results is the Richter-Sundberg formula for the local Dirichlet integral.

We study complex-symbol Toeplitz operators on weighted Fock spaces. The weight has a uniformly positive and bounded real Hessian. Local analytic distance is unchanged when an entire function is added to the symbol. We combine this distance with a complex ball average or the Berezin transform to control the local size of the symbol. Under a matching Lorentz condition on the distance, we characterize membership in every Schatten–Lorentz class. The local symbol exponent can be any $1\le q<\infty$ and is independent of the two Schatten–Lorentz indices. The proof uses positive Toeplitz estimates, Hankel estimates, and a local decomposition that preserves the initial domain. Two examples show why the local exponent and the secondary Lorentz index must be kept. We also obtain mixed mapping criteria and regular singular-value decay estimates. An error operator with a lower-order counting function does not change the leading coefficient. This gives explicit Weyl constants for quadratic weights. A radial example shows that uniform Hessian bounds alone do not imply such a constant.

Let $\mathcal{M}$ be a finite von Neumann algebra with normalized center-valued trace $T_{\mathcal{M}}$. We prove that every $A\in\mathcal{M}$ with $T_{\mathcal{M}}(A)=0$ admits a representation $A=BC-CB$, where $B,C\in\mathcal{M}$ satisfy $\|B\|\|C\|\le K\|A\|$ for an absolute constant $K$ independent of $\mathcal{M}$.

We continue to develop a theory of (Besov and) Triebel–Lizorkin spaces associated with weighted subcoercive operators on a real connected Lie group, focusing on the Triebel–Lizorkin space $F^{\infty,q}_\alpha(\beta)$, which requires a more technical treatment.

We continue to develop a theory of Besov and Triebel-Lizorkin spaces associated with weighted subcoercive operators on a real connected Lie group, focusing on the algebra properties, pointwise multipliers, localized norms, and characterization by differences.

We prove that the closed subspace of $L^1[0,1]$ spanned by the even (respectively, odd) levels of the Haar system is not isomorphic to $L^1[0,1]$.

We present a systematic study of $a$-Davis–Wielandt radius estimates for elements in a unital $C^*$-algebra through an Orlicz function approach. We derive novel bounds for the algebraic $a$-Davis–Wielandt radius and further obtain estimates for the classical algebraic Davis–Wielandt radius via polar decomposition and the Moore–Penrose inverse. The results obtained in this work extend, generalize, and unify a variety of established inequalities in the literature as special cases.

We give an explicit example of a $W^{*}$-bundle $M$ over a compact, metrizable space $K$, which has each fiber $M_{p}$ a $\textrm{II}_{1}$-factor with separable predual, and which satisfies the following negation of the generator problem: given any finite family $a_{1},\cdots,a_{n}\in M$ of continuous sections, there is a $p\in K$ (depending upon that family) so that $a_{1,p},\cdots,a_{n,p}$ do not generate $M_{p}$ as a von Neumann algebra. More generally, if $N$ is a sub-bundle of $M$ with the property that each fiber is hyperfinite, or has a Cartan, or is generated by two commuting diffuse subalgebras, or has diffuse central sequence algebra, or is generated by a single sequential commutation orbit, then given any finite family $a_{1},\cdots,a_{n}\in M$ of continuous sections, there is a $p\in K$ (depending upon that family) so that $a_{1,p},\cdots,a_{n,p}$ together with $N_{p}$ do not generate $M_{p}$. We discuss implications for the generator problem for von Neumann algebras: e.g. there is no “continuous" way to take countably many generators for a tracial von Neumann algebra with separable predual and produce a single generator, at least if such a procedure works for all von Neumann algebras simultaneously.

In their previous work devoted to the distribution of rational points near Brownian motion, the authors conjectured the existence of an oscillation principle governing the asymptotic behavior of the number of rational points with bounded denomi\-nators near the graph of a monofractal curve. In this note, a weaker form of this conjecture is shown to hold for a broad class of deterministic fractal curves. These include the classical Takagi and Weierstrass nowhere differentiable functions, and indeed a prevalent (i.e.~"large") class of functions among those which are Hölder continuous with a given exponent of regularity. This constitutes the first instance of deterministic fractal curves for which a precise count of the rational points under consideration is established.

The completely bounded Bohnenblust–Hille inequality of Arunachalam, Dutt, Escudero Gutiérrez and Palazuelos controls coefficient summability with optimal constant one, independently of the ambient dimension. We classify all nonzero equality cases over both the real and complex fields: they are precisely scalar multiples of Hadamard chains supported on equal-sided Cartesian boxes. Thus equality determines both the support and the factorization of the coefficient tensor. For each fixed degree $d\ge2$, we also establish sharp structural stability. A form with completely bounded norm one and deficit $\varepsilon$ lies within $C_d\sqrt\varepsilon$, in the critical coefficient $\ell_{2d/(d+1)}$ norm, of both a flat unimodular path and a normalized unitary chain on the same box. The corresponding unimodular and scaled unitary edges satisfy a matching normalized Frobenius estimate. At zero deficit both approximants coincide with the original Hadamard chain. All constants are independent of the ambient dimension, and the exponent $1/2$ is optimal for both approximations over either field.

We prove that, for a compact Hausdorff space $K$, the real Banach space $C(K)$ has the ball fixed point property if and only if $K$ is extremally disconnected. The new implication is obtained by constructing a fixed-point-free nonexpansive self-map of the whole closed unit ball whenever $K$ is a compact $F$-space which is not extremally disconnected. The construction uses a transfinite contractive propagation system with two increasing profiles. A successor–limit delay preserves their tail constraint and has no subsolution in the profile domain. Analysis records boundary deficits; positive operators on ordinal $C_0$-spaces and a common center determined by the two tails yield a nonexpansive synthesis satisfying an order domination inequality. The required topological families are obtained from a gap in a maximal Boolean chain in the zero-dimensional case, and from a transfinite extension of signs otherwise. The argument works in ZFC and resolves the question posed by Avilés, Japón, Lennard, Martínez-Cervantes, and Stawski.

This paper provides a complex-analytic interpretation of certain topologies on the tensor algebra, which are a central tool within strict deformation quantization. We prove that the symmetric tensor algebra $\operatorname{S}(V)$ of a barelled nuclear DF-space $V$ endowed with any of these topologies may be understood as an algebra of bounded Fréchet holomorphic functions on the strong dual space $V'_\beta$. The coarsest of the topologies then corresponds to the topology of uniform convergence on bounded subsets. The finer ones provide an infinite-dimensional generalization of the order of an entire holomorphic function. To facilitate these results, we study the interplay between boundedness and continuity of holomorphic mappings between locally convex spaces. As a byproduct, we establish a simple sufficient criterion for Fréchet holomorphy as well as several completeness results for spaces of bounded functions.

We prove that a bounded operator $T$ on a complex Hilbert space is self-adjoint whenever $|T|\leq|\re T|$, as conjectured by Fong and Tsui \cite{FT}.

We completely classify unitary 2d CFTs with central charge $c=1$ and discrete spectrum. We confirm the folklore that the only such theories are the compact free boson, its $\mathbb{Z}_2$-orbifold, and three exceptional theories obtained as orbifolds of the SU(2)$_1$ WZW model. The proof is by recasting the $c=1$ modular bootstrap as a crystalline measure problem for the density of states, and analytically solving it. We also show that, once the density of states is specified, the full CFT can be uniquely reconstructed.

We develop a theory of Sobolev, Besov and Triebel–Lizorkin spaces associated with weighted subcoercive operators on any real connected Lie group.