Papers

Let $\mathcal{T}_{n,m} = \mathcal T_n(M_m(\mathbb{C}))$ denote the operator system of all block Toeplitz matrices $ T = (( T_{i-j}))_{i,j=1}^n$ with entries $T_k \in M_m(\mathbb C) $ % T_{k} = \left[ t^{(k)}_{p-q} \right]_{p,q=1}^m. \[ T = \begin{pmatrix} T_0 & T_{-1} & \cdots & T_{-(n-1)} T_1 & T_0 & \cdots & T_{-(n-2)} \vdots & \vdots & \ddots & \vdots T_{n-1} & T_{n-2}& \cdots & T_0 \end{pmatrix} \in M_{mn}(\mathbb{C}). \] We characterize all pure unital completely positive ({\it{ucp}}) maps from $\mathcal{T}_{n,m}$ to $M_m(\mathbb{C})$. Working through the Stinespring isometry $V = (V_1, \ldots, V_n)^t \colon \mathbb{C}^m \to \mathbb{C}^{mn}$ and the matrix-valued polynomial $Q_V(z) = \sum_{i=1}^n z^{n-i} V_i$, we prove that $\varphi$ is pure if and only if it admits a unique pure \ucp extension to $M_{mn}(\mathbb{C})$ if and only if $Q_V$ has degree $n-1$ with all its roots on the unit circle $\T$. Every such pure $\varphi$ induces a \ucp map $\Phi_{Q_V}$ on $C(\mathbb{T}, M_m(\mathbb{C}))$ given by \[ \Phi_{Q_V}(f) = \int_{\mathbb{T}} Q_V(z)^* f(z) Q_V(z)\, dz. \] Let $\mathcal{Y}_m$ be the compact convex set of all \ucp maps from $C(\mathbb{T}, M_m(\mathbb{C}))$ to $M_m(\mathbb{C})$. Endowing $\mathcal{Y}_m$ with the matricial Monge-Kantorovich metric $\rho$, via an analysis of the extreme points of $\mathcal{Y}_m$ together with a point-splitting lemma, we show that the induced maps as above are $\rho$-dense in $\mathcal{Y}_m$. Consequently, if $\Bmn$ denotes the set of normalized $\Phi_{Q_V}$ where $Q_V$ is as above, then the Hausdorff distance $d_H(\Bmn, \mathcal{Y}_m) \to 0$ as $n \to \infty$, extending known results of approximation of positive regular Borel measures on the unit circle to the setting of matrix-valued completely positive maps.

We investigate the Gabor frame problem for a rational window $$g(x)= \frac{x}{(x^2+1)(x^2+4)}.$$ This window falls outside the classes covered by existing frame set characterizations for rational Herglotz functions and ratios of exponential polynomials. For rational densities ($\alpha\beta = p/q$), we leverage the Zak transform to map the Gabor frame condition directly to a finite-dimensional equivalence criterion, which is completely determined by the rank of an associated polynomial matrix. Applying this framework, we establish new frame results for every rational density $$\alpha\beta=\frac{p}{q}\in \left(0,\frac{1}{2}\right)\cup\left(\frac{1}{2},\frac{2}{3}\right)\cup \left(\frac{2}{3},\frac{5}{7}\right].$$

Classical sumset inequalities originating in additive combinatorics admit geometric analogues for convex bodies in finite-dimensional real vector spaces, as recently developed by Fradelizi and two of the authors. We develop integral analogues for geometric $\alpha$-concave functions under $\alpha$-convolutions, an operation that arises naturally in the “geometrization of probability” program. In particular, we establish a Plünnecke–Ruzsa-type inequality, as well as sharp analogues of sum-difference and Ruzsa triangle inequalities, for $\alpha$-convolutions of $\alpha$-concave functions. We also prove a sharp Rogers–Shephard-type inequality and characterize its equality cases for $\alpha$-concave functions, bridging the log-concave case studied by Alonso-Gutiérrez, González-Merino, Jiménez, and Villa and the quasi-concave case studied by Colesanti.

We prove a finite-point version of the Schwarz-Pick inequality for holomorphic self-maps of the disk. The estimate compares Bergman Gram matrices built from several points and their images, and the sharp constant depends on both the number of points and, for finite Blaschke products, the degree of the map. These matrices also arise from the Szegő-kernel metric on the symmetrized polydisc, where the estimate becomes a sharp Schwarz-Pick inequality for the induced holomorphic maps.

Let $(S^1,\delta)$ be the unit circle endowed with the arc length metric. This paper concerns the question of which subsets of $(S^1,\delta)$ can be embedded isometrically into the sequence space $\ell_1$. It is well known that every finite subset of $S^1$ admits such an embedding, but the situation for infinite subsets, even including the whole circle, has perhaps been obscured by conflicting terminology in the literature. It is worth noting that the circle avoids the classical obstructions to isometric embeddability, since it is both of negative type and hypermetric. In this paper we show that if $X \subseteq S^{1}$ is a closed set and the Lebesgue measure of $X \cap X^{\ast}$ is positive, where $X^{\ast}$ is the antipodal set of $X$, then it is impossible to isometrically embed $(X, \delta)$ in $\ell_{1}$. As a result, no subset of $S^{1}$ with Lebesgue measure greater than $\pi$ can be isometrically embedded in $\ell_1$. Conversely, we show that any closed subset of $S^1$ whose intersection with any half-circle has measure zero can be isometrically embedded in $\ell_1$. These results have several consequences. They imply that the classical Banach space $L_1[0, 1]$, considered purely as a metric space, does not isometrically embed in $\ell_{1}$. Secondly, they yield a simple proof that a metric graph $(M, d)$ embeds isometrically in $\ell_1$ if and only if it is a tree. In contrast to $S^{1}$, we show that some related metric spaces, such as the cylinder and the flat torus, contain finite subsets that cannot be isometrically embedded in $\ell_1$.

We prove a Kolmogorov–Riesz compactness theorem for Banach lattices over spaces of homogeneous type. Especially, we consider the one for $L^1$. Our characterization is formulated in terms of boundedness, tightness, and approximation by averaging operators, which replace translations in the absence of a group structure. The result requires neither the Vitali covering property nor continuity assumptions on the measure of balls, lower volume bounds, or finiteness of the underlying measure space, thereby extending several recent compactness criteria in the literature. Our approach begins with a new proof of the Kolmogorov–Riesz theorem in $L^1$, which avoids reflexivity and yields a unified treatment of $L^p$-spaces for $1\le p<\infty$. We then establish an approximation theorem based on Christ's dyadic cubes and use it to characterize the closure of compactly supported continuous functions in general ball Banach function spaces. Applications include compactness criteria for rearrangement-invariant Banach function spaces and Morrey spaces. In the Euclidean setting, we also identify the resulting approximation space with the classical heat-semigroup (equivalently, mollifier) closure.

We develop a diffeological framework for the geometry of Euler–Reynolds subsolutions of the incompressible Euler equations. Passing to a lifted formulation in which the velocity, quadratic flux, and trace-free Reynolds stress are treated as independent variables, we construct a diffeological limit space obtained as the weak closure of smooth strict subsolutions. Its internal tangent spaces provide an intrinsic notion of infinitesimal deformation despite the absence of any underlying manifold structure. We prove that ambient realizations of internal tangent vectors satisfy the linearized Euler–Reynolds equations in the sense of distributions, thereby establishing a natural first-order theory for lifted weak limit spaces. We further describe the tangent directions compatible with the Euler locus and identify the kernels of the natural velocity, flux, and full-state observables. These results distinguish observable perturbations from hidden stress-gauge directions that encode infinitesimal variations of the Reynolds stress. Finally, we introduce a finite-mixture model for lifted Euler–Reynolds states whose differential realizes explicit tangent directions and relates Reynolds stress to infinitesimal phase splitting. Under a genericity assumption, every deviatoric stress tensor is realized by such a first-order mixture defect, yielding phase-counting bounds and a minimality result for the observable hierarchy.

Recently, A. Tyulenev and the author studied the class of metric spaces $\mathcal{X}$ such that for every mapping $\gamma \colon [0,1]\to\mathcal{X}$, there exists a $1$-Lipschitz function $F\colon \mathcal{X}\to \mathbb{R}$ that catches the total variation of $\gamma$, i.e. such that $\operatorname{V}_{F\circ\gamma} \gtrsim \operatorname{V}_\gamma$. In this short note, we show that a metric space $\mathcal{X}$ enjoys this property if and only if there exists a random $1$-Lipschitz function $f$ on $\mathcal{X}$ such that $\mathbb{E} |f(x) - f(y)| \gtrsim \rho(x,y)$ for every $x$ and $y$ in $\mathcal{X}$.

For $ n \geq 2$, $A\in M_n(\mathbb C)$ and $0<|q|\leq 1$, let $\Omega_q(A)=q^{-1}W_q(A)$ be the scaled $q$-numerical range. We prove that for every $\gamma \geq 1$, \[ \Omega_{\eta(\gamma)}(A) =\bigcup_{\kappa(S)\leq\gamma}W(S^{-1}AS), \qquad \eta(\gamma)=\frac{2}{\gamma+\gamma^{-1}}, \] where $\kappa(S)=\|S\|\,\|S^{-1}\|$. As a consequence, we prove the sharp inequality \[ \|p(A)\|\leq \max\!\left\{1,\frac{2|q|}{1+\sqrt{1-|q|^2}}\right\} \max_{z\in\Omega_q(A)}|p(z)|, \] for all polynomials $p$.

We study the growth of powers of mean-L-stable operators on Banach spaces. We show that, for linear operators, mean-L-stability is equivalent to uniform boundedness in density; this yields $\|T^n\|=O(n)$ on every Banach space. On Hilbert spaces we prove that mean-L-stability is equivalent to absolute Cesàro boundedness and obtain $\|T^n\|=O(n^{1/2-c_T})$ for some $c_T>0$. For positive mean-L-stable operators on abstract $L^p$-spaces, $1\le p<\infty$, we similarly obtain $\|T^n\|=O(n^{1/p-c_T})$, where in both cases the positive constant $c_T$ cannot be chosen uniformly over all such operators. For positive mean-L-stable operators on $p$-convex Banach lattices, we prove the bound $O(n^{1/p})$ and construct positive topologically mixing operators $T_p$ for which $\|T_p^n\|\asymp n^{1/p}$, where $1\leq p<\infty$. These operators satisfy a uniform weak $(p,p)$ orbit estimate, while the averages of $\|T_p^nx\|^s$ are bounded for $s<p$, of order $\log N$ for $s=p$, and of order $N^{s/p-1}$ for $s>p$. The operator $T_1$ is uniformly Kreiss bounded and has linear power growth, answering a question of Montes-Rodríguez, Sánchez-Álvarez and Zemánek (2005). Moreover, $T_1$ is mean-L-stable and mean Li–Yorke chaotic, while it is not distributionally chaotic. This answers a question of Bernardes, Bonilla and Peris (2020).

We propose Choquet-type extensions for set-pair functions, an $L^p$-integration of a family of such extensions, and the associated min-max extension constants, and apply these to investigate variational problems and establish sharp bounds for many combinatorial and geometric quantities. Specifically, we use Choquet-type extensions to construct equivalent functional representations of fractional optimization problems with set-variables, which find applications in various settings, such as maxcut and bipartiteness ratio on graph limits, and Cheeger constants on Riemannian manifolds. We further establish the $L^p$ integration of a family of Choquet-type extensions, and apply it to derive spectral bounds for conductance and other combinatorial quantities on measure spaces. A monotonicity inequality on min-max extension constants is proposed, which unifies classical estimates and uncovers new bounds for many geometric and combinatorial quantities, such as torsional rigidity, Cheeger constants, Dirichlet $p$-isoperimetric constant, and $p$-Laplacian eigenvalues, in seemingly unrelated underlying structures—-including hypergraphs, graph limits, Riemannian manifolds and metric measure spaces.

Let $\psi({\mathbf x})$, ${\mathbf x} \in{\mathbb R}^{3N}$, be an eigenfunction of the $N$-particle atomic Schrödinger operator. We consider the one-particle density matrix $\gamma(x, y)$ and one-particle kinetic energy density $\varkappa(x, y)$, $x, y\in {\mathbb R}^3$, associated with the eigenfunction $\psi$. Both functions play a central role in quantum chemistry computations of atomic and molecular bound states: the knowledge of the eigenvalue behaviour of the integral operators ${\sf{\Gamma}}$ and ${\sf{K}}$ with kernels $\gamma(x, y)$ and $\varkappa(x, y)$ serves to estimate the errors due to finite-dimensional approximations. We find the following asymptotic formulas for their eigenvalues $\lambda_k({\sf{\Gamma}})>0$ and $\lambda_k({\sf{K}})>0$: \[ \lim_{k\to \infty} k^{\frac{8}{3}} \,\lambda_k({\sf{\Gamma}}) = A^{\frac{8}{3}},\quad \lim_{k\to \infty} k^2\,\lambda_k({\sf{K}}) = B^2, \] where $A$ and $B$ are non-negative constants given explicitly in terms of the eigenfunction $\psi$. These asymptotics are determined by the singularities of the function $\psi$ at pair coalescence points of the particles. To identify and isolate these singularities we use some recent regularity results for $\psi$. At the last step we apply Birman-Solomyak spectral asymptotics results for pseudodifferential operators with homogeneous symbols. In the special case where the eigenfunction $\psi$ is totally antisymmetric, it exhibits enhanced regularity, which leads to a faster decay of the eigenvalues $\lambda_k(\sf{\Gamma})$ and $\lambda_k(\sf{K})$. The asymptotic formulas take the form \[ \lim_{k\to \infty} k^{\frac{10}{3}} \,\lambda_k({\sf\Gamma}) = \big(A_{asym}\big)^{\frac{10}{3}},\quad \lim_{k\to \infty} k^{\frac{8}{3}} \,\lambda_k({\sf K}) = \big(B_{asym}\big)^{\frac{8}{3}}, \] where $A_{asym}$ and $B_{asym}$ are non-negative constants given explicitly in terms of the gradient of $\psi$.

In previous work, the density of the martingale limit in Schröder branching processes was expressed as a convergent double power-law series with oscillatory terms. The proof relied on two assumptions, one of which imposed a geometric restriction on the critical angle of the Julia set of the offspring generating function near $1$. In this paper, we show that both assumptions can be removed. We derive explicit bounds on the expansion coefficients, which imply locally uniform convergence of the double series. The coefficients decay exponentially in one summation index, with the rate explicitly determined by the critical angle, and super-exponentially in the other. Finally, we investigate the behavior of the critical angle and describe regimes in which it approaches its minimum. This analysis shows how the geometry of the Julia set influences the magnitude of the oscillatory corrections in the left-tail expansion.

Let $T$ be an exponentially stable strongly continuous semigroup on a Hilbert space $X$, $A$ its generator, $X_1$ the domain of $A$ with the norm $\|v\|_1 := \|Av\|$, and $C$ an admissible observation operator taking values in a Hilbert space $Y$. Russell and Weiss conjectured that if $(A,C)$ satisfies the infinite-dimensional Hautus test, then $(A,C)$ is exactly observable. After a counterexample was found their conjecture was modified to include the additional assumption that $T$ is similar to a contraction semigroup. We disprove this modified conjecture by constructing a counterexample in which $A$ is normal, $X$ has an orthonormal basis of eigenvectors of $A$ and $C: X_1 \to Y$ is Hilbert-Schmidt. Since $A$ is normal and $T$ is exponentially stable, it follows that $T$ is a contraction semigroup. In contrast, we prove that the Hautus test implies exact observability for self-adjoint $A$. Finally, we show that if $A$ is normal and $C:X_1 \to Y$ is compact, then the Hautus test implies that $A$ has compact resolvent. Consequently, for normal $A$ and finite-dimensional $Y$ the Hautus test implies exact observability.

In this paper, we study the boundedness properties of Erdélyi–Kober fractional integrals and Mellin fractional integrals on weighted Lebesgue spaces over $% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion _{+}=$ $\left( 0,\infty \right) $. We establish sufficient conditions on different weight functions to ensure the boundedness of these operators between weighted integrable spaces. Our approach is mainly based on weighted Hardy-type inequalities, Hölder estimates, and suitable changes of variables associated with the multiplicative structure of the operators. We first investigate a class of Erdélyi–Kober type integral operators and derive weighted $L^{p}$-inequalities under appropriate assumptions on the weights. These results extend several classical inequalities related to Hardy operators and fractional integrals. We then consider Mellin fractional integral operators and obtain analogous boundedness results in weighted Lebesgue spaces. The obtained estimates reveal a close connection between Erd% élyi–Kober operators and Mellin-type fractional integrals within the framework of multiplicative harmonic analysis. The results presented in this paper provide a unified treatment of these fractional integral operators in weighted settings and generalize various previously known boundedness results. In particular, our conditions on the weights characterize the continuity of the operators on weighted $L^{p}$-spaces and illustrate the role played by the multiplicative structure of the underlying measure space.

We define Sobolev spaces associated with the Grushin operator with or without drift. We study some of their norm equivalences and establish embedding and algebra properties. We also prove the boundedness of the Riesz transforms on these spaces.

In this note we provide a simple explicit example of a discrete set $\Lambda\subset\mathbb{R}$ of unbounded density such that the Fourier transform of its counting measure $\widehat{\delta}_\Lambda$ is a tempered distribution of the form $c\delta_0'' + \mu$ where $\mu$ is a discrete measure supported by a union of three arithmetic progressions. This example is then used to construct a translational tiling of unbounded density of the real line by a function with compact support.

In this article we study the spectral problem related to the fractional Laplace-Beltrami equation on $(\mathbb{R}^d,g)$ and establish its equivalence with the classical anisotropic Helmholtz equation. The proof is based on Seeley's construction of complex powers of elliptic operators and the pseudodifferential symbolic calculus. As an application, we describe the related fixed-frequency inverse scattering problem of recovering the metric from the scattering amplitude.

In recent years, several generalizations of the numerical radius for bounded linear operators on Hilbert spaces have been introduced and extensively studied. These generalizations provide refined tools for investigating operator inequalities and spectral properties. In this paper, we investigate several generalized numerical radii and establish relationships among them. We also establish several relationships among these generalized numerical radii and show that the $(s,t)$-weighted numerical radius can be represented as a rescaled form of the $t$-weighted numerical radius.

For every $n\geq1$ and $\gamma>-1$, we construct $f\in L^1(\mathbb B_n,dv_\gamma)$ whose Toeplitz form extends to a bounded, noncompact operator on the weighted Bergman spaces $A^2_\gamma(\mathbb B_n)$ and whose Berezin transform vanishes at the boundary. Boundary vanishing of the Berezin transform therefore does not imply compactness for Toeplitz operators with integrable symbols, in contrast with operators in the Toeplitz algebra generated by bounded symbols; this answers a question of Bauer and Isralowitz. The construction approximates rank-one operators in norm by Toeplitz operators with smooth, compactly supported symbols and transports suitably separated blocks toward the boundary. On the unweighted disk, a diagonal version produces a real symbol with the same vanishing and noncompactness properties, for which the operator is positive and Zorboska's two-sided localization condition holds at $p=3$. Her hypothesis $p>3$ is shown to be sharp.

In this paper we develop quantitative approximation results for shallow neural networks constructed using a dictionary based on metaplectic operators. First, we extend the concept of Barron spaces by considering a symplectically motivated extension of the Fourier transform, known as the metaplectic transform. Then, after establishing embedding between metaplectic Barron spaces and Sobolev spaces we consider a neural metaplectic dictionary and we prove Monte-Carlo approximation bounds for metaplectic Barron functions using finite linear combinations of atoms of the dictionary. Finally, we validate the introduction of the neural metaplectic dictionary by devising a deep neural network architecture that uses as building blocks the atoms of the dictionary. We test it to approximate solutions of time-dependent Schrödinger equations, demonstrating better performance compared to classical phyisics informed neural networks architectures.

For $0 \le \alpha \le 1$ and $-1 \le r < 0$, let $\#_{r,\alpha}$ denote the weighted Kubo-Ando power mean of order $r$ and weight $\alpha$. Ando proved that, for every unitarily invariant norm, $\| A \ \#_{r,\alpha}\ B \| \le \| A \|\ \#_{r,\alpha}\ \| B \|$. We show that this inequality remains valid also for the Schatten $q$-quasi-norms with $0 < q < 1$. More generally, our main result resolves a problem recently posed by Hiai and Lim in the stronger form of weak log-majorization for multivariable Lim-Pálfia power means of negative order. It yields analogous Schatten quasi-norm inequalities for every finite family of positive definite matrices.

We extend a finite-support $q$-Hermitian radial calculus to Clifford-valued coordinate polynomials. For $N$ bosonic Hermitian vector labels and finite fermionic support, the stable range $m\ge 2N$ admits a canonical Gram–harmonic normal form by Howe separation. A label–Capelli inverse normalizes the Clifford contractions, while a finite Wick–Chevalley conjugation lifts the divided-power scalar gauge to the full Clifford PBW module. Coupling the normalized contractions to contractible fermionic seed complexes gives two conjugate polynomial-preserving $q$-Hermitian coordinate families. They are square-zero, anticommute within each polarization, restrict exactly to the radial PBW calculus, have scalar boundary trace $[2m-2n]_q$, and recover the classical Hermitian super Dirac pair as $q\to1$. For the mixed polarizations we introduce a relative triangular transport. The Grassmann relations for the normalized Capelli contractions hold on the full coordinate module and yield a labelwise factorization. Hence all higher-filtration mixed terms are finite subset products of explicit local defects, and the mixed polarization algebra closes for arbitrary finite label sets and finite fermionic support. The classical complex structure is common to both polarizations, whereas the two full transported Clifford–Weyl products are distinct for $0<q<1$. We also show that the fermionic Cartan denominator is forced in the natural polynomial first-order Berezin–Weyl class and determine the minimal factorwise Ore localization needed for the conjugate all-charge homotopies. The $q$-dependent Cartan orbit is nonresonant for $m\ge\max\{2N,n\}$. Finally, fixed finite-order differential and finite nonzero-shift realizations on the undeformed coordinate algebra are ruled out.

Let $\lambda_{\mathbb K}(m)$ denote the maximal absolute projection constant among $m$-dimensional Banach spaces over $\mathbb K=\mathbb R$ or $\mathbb C$. Its exact value is known only in a few cases, and determining it remains a challenging problem. In this note, we investigate several structured vector configurations that naturally arise in this context. We first recall the connection between maximal projection constants and tight frames, then consider biangular tight frames, and show that their relative and quasimaximal projection constants coincide. A particularly interesting example is provided by the midpoints of the edges of a regular simplex, which yield natural lower bounds for $\lambda_{\mathbb R}(m)$. Numerical evidence suggests that these bounds may be sharp in dimensions $6$ and $8$. We also discuss weighted spherical $(2,2)$-designs with a small number of vectors and their connection with maximal projection constants. Examples in dimensions $4$ and $5$ indicate that the sign patterns of their Gram matrices may play an important role. These observations lead us to formulate several conjectures concerning maximal absolute projection constants and the vector configurations associated with them.

We prove the existence of optimizers for a class of homogeneous Gagliardo-Nirenberg inequalities of the form $$ \|D^{s}\phi\|_{L^q} \leq C \|D^{s_1}\phi\|_{L^p}^{1-\theta} \|D^{s_2}\phi\|_{L^2}^{\theta}, \quad s_1<s_2, $$ under the scaling relation $$ \frac{1}{q}-\frac{s}{d} = (1-\theta)\left(\frac{1}{p}-\frac{s_1}{d}\right) + \theta\left(\frac{1}{2}-\frac{s_2}{d}\right), $$ and the strict interpolation condition $$ s<(1-\theta)s_1+\theta s_2. $$ When $s>s_1$, the problem reduces, after shifting the derivative orders, to the result of Bellazzini, Frank and Visciglia, whereas the case $s\leq s_1$ follows from the present paper. As an application, we study normalized solutions of the associated Euler-Lagrange equations under the constraint $$ \|D^{s_1}\psi\|_{L^p}^p=\lambda>0. $$ In particular, the case $p=2$ includes the stationary equation arising from the Majda-McLaughlin-Tabak (MMT) model.

In this paper, we consider the notion of the weighted numerical radius, denoted by $ \omega_{\nu}(A)$, for the class of accretive matrices, where \\ $ \nu \in [0, 1].$ This notion generalizes both the classical numerical radius and the operator norm. All of the obtained inequalities reduce to the corresponding classical inequalities when $ \nu=1/2 $ or when the matrix is positive definite.

The (B)-theorem of Cordero-Erausquin, Fradelizi and Maurey states that if $\gamma$ is the standard Gaussian in $\mathbb R^n$, $K \subset \mathbb R^n$ is an origin-symmetric convex set, and $s, t \in \mathbb R$ then $\gamma\left(e^{\frac{s + t}{2}} K\right) \ge \sqrt{\gamma(e^{s} K) \gamma(e^{t} K)}$. Herscovici, Livshyts, Rotem and Volberg proved a stability version of this result, showing that if one has equality up to a factor $(1 + \epsilon)$ in the (B)-inequality for $K$ then the inradius of $K$ must be either “very large” or “very small,” where the bounds depend on $\epsilon$ and on $n$. We give a new stability estimate which is dimension-free and also yields more precise information about bodies which are near-optimizers of the (B)-inequality. In particular, our results imply that if $\gamma\left(e^{\frac{s + t}{2}} K\right) \le (1 + \epsilon) \sqrt{\gamma(e^{s} K) \gamma(e^{t} K)}$, then every principal component of the covariance matrix of the probability measure obtained by restricting the Gaussian to $K$ must either be at least $1 - O(\epsilon)$ or at most $O(\epsilon)$, which is sharp. Our method extends immediately to yield stability estimates for generalizations of the (B)-inequality, namely the “strong” and “functional” (B)-inequalities, which reduce to spectral questions about $1$-log-concave measures on $\mathbb R^n$.

Up to now operators with uncomplemented null-space have been playing only a minor role in regularization theory for ill-posed inverse problems in Banach spaces. We show that ill-posedness due to an uncomplemented null-space and ill-posedness due to a non-closed range can be regarded as dual concepts and we may switch between both views at will by restricting or extending the ill-posed operator under consideration in a suitable well-defined way. We apply the duality result to Mazur-type operators, which are a class of bounded linear operators with uncomplemented null-space. Mazur-type operators in their natural form are not accessible to Tikhonov-type regularization, but after transfering ill-posedness to the range Tikhonov regularization can be applied. On the other hand, the null-space view of ill-posedness may open up an alternative path for developing new regularization methods for operators ill-posed in the classical sense, that is, for operators ill-posed due to a non-closed range.

The finite-generator corona theorem for bounded slice regular functions on the quaternionic unit ball was recently established by Colombo, Pozzi, Sabadini, and Wick. In the present paper, we extend the quaternionic \(H^\infty\)-corona theorem to countably many generators and obtain quantitative estimates that are independent of the cardinality of the generating family. We also establish the corresponding \(H^p\)-corona theorem for the full range \(1\leq p<\infty\), with quantitative norm estimates for both finite and countable families of generators. In the Hilbert-space setting, we prove a quaternionic Leech factorization theorem for Hardy-space multipliers and derive, as a consequence, a Toeplitz corona characterization. Our approach is based on a fixed-slice \(2\times2\) complex matrix realization of the slice regular product, together with operator-valued corona and factorization techniques.

Let $A$ be a unital $C^*$-algebra and $I$ be a symmetrically normed ideal of $A$. I introduce and study lexicographic functional calculus (LFC), a kind of multivariate functional calculus for tuples $(a_1,\ldots,a_m)$ of noncommuting self-adjoint elements of $A$ ”acting in lexicographic order,” i.e., from left to right, with an element $b_i \in I$ ”inserted” between the action of $a_i$ and $a_{i+1}$ for each $i=1,\ldots,m-1$. The reason for its introduction is an application to ”functional calculus calculus,” the differential calculus of maps induced by single-variate (continuous) functional calculus. Specifically, I prove that if $f\colon\mathbb{R}\to\mathbb{C}$ is sufficiently regular and $a\in A_{\mathrm{sa}}:=\{c\in A:c^*=c\}$, then $f_{a,I}(b):=f(a+b)-f(a)\in I$ for all $b\in I_{\mathrm{sa}}:=I\cap A_{\mathrm{sa}}$, the map $f_{a,I}\colon I_{\mathrm{sa}}\to I$ is Fréchet $C^k$, and the $k^{\text{th}}$ Fréchet derivative of $f_{a,I}$ may be written in terms of LFC applied to the $k^{\text{th}}$ divided difference of $f$, a function of $k+1$ variables. This result recovers or vastly generalizes nearly all comparable results in the literature. For example, it simultaneously recovers the following three highly related results on the regularity of the function $f_A\colon A_{\mathrm{sa}}\to A$ defined by $a\mapsto f(a)$: (1) If $A$ is commutative and $f\in C^k(\mathbb{R})$, then $f_A$ is Fréchet $C^k$; (2) if $A$ is finite dimensional and $f\in C^k(\mathbb{R})$, then $f_A$ is Fréchet $C^k$; and (3) if $f\colon\mathbb{R}\to\mathbb{C}$ is ”slightly better than $C^k$,” e.g., belongs to the homogeneous Besov space $\dot{B}_1^{k,\infty}(\mathbb{R})$, then $f_A$ is Fréchet $C^k$ no matter the choice of $A$. Before LFC, there was no unifying framework for results (1)–(3); in particular, there was no single result of which they were all corollaries.