The unit ball $B_p^n(\mathbb{R})$ of the finite-dimensional Schatten trace class $\mathcal S_p^n$ consists of all real $n\times n$ matrices $A$ whose singular values $s_1(A),\ldots,s_n(A)$ satisfy $s_1^p(A)+\ldots+s_n^p(A)\leq 1$, where $p>0$. Saint Raymond [Studia Math.\ 80, 63--75, 1984] showed that the limit $$ \lim_{n\to\infty} n^{1/2 + 1/p} \big(\text{Vol}\, B_p^n(\mathbb{R})\big)^{1/n^2} $$ exists in $(0,\infty)$ and provided both lower and upper bounds. In this paper we determine the precise limiting constant based on ideas from the theory of logarithmic potentials with external fields. A similar result is obtained for complex Schatten balls. As an application we compute the precise asymptotic volume ratio of the Schatten $p$-balls, as $n\to\infty$, thereby extending Saint Raymond's estimate in the case of the nuclear norm ($p=1$) to the full regime $1\leq p \leq \infty$ with exact limiting behavior.
We study the first and second orders of the asymptotic expansion, as the dimension goes to infinity, of the moments of the Hilbert-Schmidt norm of a uniformly distributed matrix in the p-Schatten unit ball. We consider the case of matrices with real, complex or quaternionic entries, self-adjoint or not. When p > 3, this asymptotic expansion allows us to establish a generalized version of the variance conjecture for the family of p-Schatten unit balls of self-adjoint matrices.
In this paper, we characterize the class of extremal points of the unit ball of the Hessian-Schatten total variation (HTV) functional. The underlying motivation for our work stems from a general representer theorem that characterizes the solution set of regularized linear inverse problems in terms of the extremal points of the regularization ball. Our analysis is mainly based on studying the class of continuous and piecewise linear (CPWL) functions. In particular, we show that in dimension $d=2$, CPWL functions are dense in the unit ball of the HTV functional. Moreover, we prove that a CPWL fu…
The optimal 2-uniform convexity of Schatten classes $S_p, 1<p\le 2$ was first proved by Ball, Carlen and Lieb \cite{BCL94}. In this note we revisit this result using multiple operator integrals and generalized monotone metrics in quantum information theory.
In this paper we prove that the ball is a maximizer of the Schatten $p$-norm of some convolution type integral operators with non-increasing kernels among all domains of a given measure in $\mathbb R^{d}$. We also show that the equilateral triangle has the largest Schatten $p$-norm among all triangles of a given area. Some physical motivations for our results are also presented.
In this note we prove that the ball is a maximiser of some Schatten $p$-norms of the Riesz potential operators among all domains of a given measure in $\mathbb R^{d}$. In particular, the result is valid for the polyharmonic Newton potential operator, which is related to a nonlocal boundary value problem for the poly-Laplacian extending the one considered by M. Kac in the case of the Laplacian, so we obtain and isoperimetric inequalities for its eigenvalues as well, namely, analogues of Rayleigh-Faber-Krahn and Hong-Krahn-Szegö inequalities.
The boundedness and compactness of Toeplitz operator from $A_ω^p$ to $A_ω^q$ with doubling weights $ω$ are studied in this paper. The characterizations of Schatten class Toeplitz operators and Volterra operators on $A_ω^2$ are also investigated.
We study the Carleson measures and the Toeplitz operators on the class of so-called small weighted Bergman spaces, introduced recently by Seip. A characterization of Carleson measures is obtained which extends Seip's results from the unit disc of $\mathbb C$ to the unit ball of $\mathbb C^n$. We use this characterization to give necessary and sufficient conditions for the boundedness and compactness of Toeplitz operators. Finally, we study the Schatten $p$ classes membership of Toeplitz operators for $1<p<\infty$.
Let $H$ and $H'$ be a complex Hilbert spaces. For $p\in(1, \infty)\backslash\{2\}$ we consider the Banach space $C_p(H)$ of all $p$-Schatten von Neumann operators, whose unit sphere is denoted by $S(C_p(H))$. We prove that every surjective isometry $Δ: S(C_p(H))\to S(C_p(H'))$ can be extended to a complex linear or to a conjugate linear surjective isometry $T:C_p(H)\to C_p(H')$.
We study a Toeplitz type operator $Q_μ$ between the holomorphic Hardy spaces $H^p$ and $H^q$ of the unit ball. Here the generating symbol $μ$ is assumed to a positive Borel measure. This kind of operator is related to many classical mappings acting on Hardy spaces, such as composition operators, the Volterra type integration operators and Carleson embeddings. We completely characterize the boundedness and compactness of $Q_μ:H^p\to H^q$ for the full range $1<p,q<\infty$; and also describe the membership in the Schatten classes of $H^2$. In the last section of the paper, we demonstrate the usef…
We study the volume of the intersection of two unit balls from one of the classical matrix ensembles GOE, GUE and GSE, as the dimension tends to infinity. This can be regarded as a matrix analogue of a result of Schechtman and Schmuckenschläger for classical $\ell_p$-balls [Schechtman and Schmuckenschläger, GAFA Lecture Notes, 1991]. The proof of our result is based on two ingredients, which are of independent interest. The first one is a weak law of large numbers for a point chosen uniformly at random in the unit ball of such a matrix ensemble. The second one is an explicit computation of the…
We show that the Euclidean ball has the smallest volume among sublevel sets of nonnegative forms of bounded Bombieri norm as well as among sublevel sets of sum of squares forms whose Gram matrix has bounded Frobenius or nuclear (or, more generally, p-Schatten) norm. These volume-minimizing properties of the Euclidean ball with respect to its representation (as a sublevel set of a form of fixed even degree) complement its numerous intrinsic geometric properties. We also provide a probabilistic interpretation of the results.
We examine the group of isometries of the open unit ball of a complex Banach space of certain bounded linear operators equipped with the Carathéodory metric. Therein we obtain a charactrization of the normal isometries in terms of their special type of fixed points.
Let $\mathcal{D}_v$ denote the Dirichlet type space in the unit disc induced by a radial weight $v$ for which $\widehat{v}(r)=\int_r^1 v(s)\,ds$ satisfies the doubling property $\int_r^1 v(s)\,ds\le C \int_{\frac{1+r}{2}}^1 v(s)\,ds.$ In this paper, we characterize the Schatten classes $S_p(\mathcal{D}_v)$ of the generalized Hilbert operators \begin{equation*} \mathcal{H}_g(f)(z)=\int_0^1f(t)g'(tz)\,dt \end{equation*} acting on $\mathcal{D}_v$, where $v$ satisfies the Muckenhoupt-type conditions $$ \sup_{0<r<1}\left(\int_r^1 \frac{\widehat{v}(s)}{(1-s)^2} \,ds\right)^{1/2} \left(\int_0^r \frac…
We present a new blow-up method that allows for establishing the first general formula to compute the perimeter measure with respect to the spherical Hausdorff measure in noncommutative nilpotent groups. This result leads us to an unexpected relationship between the area formula with respect to a distance and the profile of its corresponding unit ball.
We prove the plasticity of the unit ball of $c$. That is, we show that every non-expansive bijection from the unit ball of $c$ onto itself is an isometry. We also demonstrate a slightly weaker property for the unit ball of $c_0$ -- we prove that a non-expansive bijection is an isometry, provided that it has a continuous inverse.
We study Banach spaces with a weak stable unit ball, that is Banach spaces where every convex combination of relatively weakly open subsets in its unit ball is again a relatively weakly open subset in its unit ball. It is proved that the class of $L_1$ preduals with a weak stable unit ball agree with those $L_1$ preduals which are purely atomic, that is preduals of $\ell_1(Γ)$ for some set $Γ$, getting in this way a complete geometrical characterization of purely atomic preduals of $L_1$, which answers a setting problem. As a consequence, we prove the equivalence for $L_1$ preduals of differen…
We show that every nonvoid relatively weakly open subset, in particular every slice, of the unit ball of an infinite-dimensional uniform algebra has diameter~2.
The aim of this paper is to characterize those linear maps from a von Neumann factor $\A$ into itself which preserve the extreme points of the unit ball of $\A$. For example, we show that if $\A$ is infinite, then every such linear preserver can be written as a fixed unitary operator times either a unital *-homomorphism or a unital *-antihomomorphism.
Extending recent results by Cascales, Kadets, Orihuela and Wingler (2016), Kadets and Zavarzina (2017), and Zavarzina (2017) we demonstrate that for every Banach space $X$ and every collection $Z_i, i\in I$ of strictly convex Banach spaces every non-expansive bijection from the unit ball of $X$ to the unit ball of sum of $Z_i$ by $\ell_1$ is an isometry.
We identify the exposed points of the unit ball of the dual space of the ball algebra. As a corollary, we show that the predual of $H^\infty(\mathbb{B}_d)$ has no extreme points in its unit ball.
We prove that Banach spaces $\ell_1\oplus_2\mathbb{R}$ and $X\oplus_\infty Y$, with strictly convex $X$ and $Y$, have plastic unit balls (we call a metric space plastic if every non-expansive bijection from this space onto itself is an isometry).
The notion of a unitary realization is used to estimate derivatives of arbitrary order of functions in the Schur-Agler class on the polydisk and unit ball.
Let $E$ be a $(\mathrm{IV})$-polyhedral Banach space. We show that, for each $ε>0$, $E$ admits an $ε$-equivalent $\mathrm{(V)}$-polyhedral norm such that the corresponding closed unit ball is the closed convex hull of its extreme points. In particular, we obtain that every separable isomorphically polyhedral Banach space, for each $ε>0$, admits an $ε$-equivalent $(\mathrm{V})$-polyhedral norm such that the corresponding closed unit ball is the closed convex hull of its extreme points.
Given a weight function, we define the Bergman type projection with values in the corresponding weighted Bergman space on the unit ball $\mathbb B_n$ of $\mathbb C^n, n>1$. We characterize the radial weights such that this projection is bounded from $L^\infty$ to the Bloch space $\mathcal B$.
As an appropriate analog of the Euclidean short-time Fourier transform, we study a windowed version of the Helgason-Fourier transform on the complex unit ball and translate the theory of modulation/coorbit spaces. As a result, atomic decompositions by means of Banach frames are obtained.
We prove that the following three properties for a Banach space are all different from each other: every finite convex combination of slices of the unit ball is (1) relatively weakly open, (2) has nonempty interior in relative weak topology of the unit ball, and (3) intersects the unit sphere. In particular, the $1$-sum of two Banach spaces does not have property (1), but it has property (2) if both the spaces have property (1); the Banach space $C[0,1]$ does not have property (2), although it has property (3).
We completely describe spaces of multipliers of certain harmonic function spaces of Bergman type in R^n.This is the first sharp result of this kind for Bergman type mixed norm spaces of harmonic functions in the unit ball of R^n
In this paper, we introduce a unitary invariant $Γ$ defined on the unit ball of $B(H)^n$ in terms of the characteristic function, the noncommutative Poisson kernel, and the defect operator associated with a row contraction. We show that $Γ$ detects the pure row isometries and completely classify them up to a unitary equivalence. We also show that $Γ$ detects the pure row contractions with polynomial characteristic functions and completely non-coisometric row contractions. In particular, we show that any completely non-coisometric row contraction with constant characteristic function is homogen…
We establish weak factorizations for a weighted Bergman space $A^p_{\a}$, with $1<p<\infty$, into two weighted Bergman spaces on the unit ball of $\C^n$. To obtain this result, we characterize bounded Hankel forms on weighted Bergman spaces on the unit ball of $\C^n$.
We characterize model polynomials that are cyclic in Dirichlet-type spaces in the unit ball of $\mathbb{C}^n$, and we give a sufficient capacity condition in order to identify non-cyclic vectors.
In this paper, a new characterization is provided for the boundedness, compactness and essential norm of the difference of two weighted composition operators on weighted-type spaces in the unit ball of $\mathbb{C}^n$.
In this paper, we study the quaternionic counterpart of complex Fock spaces $\mathfrak{F}_α^p ( 0<p<\infty$ and for some parameter $α$) of entire slice hyperholomorphic functions in an Euclidean unit ball $\mathbb{B}^n$ in $\mathbb{H}^n.$ We also study some basic properties of these spaces.
The volume of the unit ball of the Lebesgue sequence space $\ell_p^m$ is very well known since the times of Dirichlet. We calculate the volume of the unit ball in the mixed norm $\ell^n_q(\ell_p^m)$, whose special cases are nowadays popular in machine learning under the name of group lasso. We consider the real as well as the complex case. The result is given by a closed formula involving the gamma function, only slightly more complicated than the one of Dirichlet. We close by an overview of open problems.
Every analytic self-map of the unit ball of a Hilbert space induces a bounded composition operator on the space of Bloch functions. Necessary and sufficient conditions for compactness of such composition operators are provided, as well as some examples that clarify the connections among such conditions.
If F is an automorphism of the spectral unit ball, we show that, in a neighborhood of any cyclic (i.e. non-derogatory) matrix of the ball, the map F can be written as conjugation by a holomorphically varying non singular matrix. This provides a shorter proof of a theorem of J. Rostand, with a slightly stronger result.
The boundedness and compactness of weighted composition operators from $H^\infty$ to the Bloch space in the unit ball of Cn are investigated in this paper. In particular, some new characterizations for the boundedness and the essential norm of weighted composition operators are given.
We show that the unit ball of a Hilbert space in its weak topology is a continuous image of the countable power of the Alexandroff compactification of a discrete set, and we deduce some combinatorial properties of its lattice of open sets which are not shared by the balls of other equivalent norms when the space is nonseparable.
In this work, we prove that if S is a dual bounded sequence in the unit ball B of C^n for the Hardy class H^p(B), then S is H^s(B) interpolating with the linear extension property.
We obtain a formula that relates the spherical moments of the multiplication tuple on a Dirichlet-type space to a complex moment problem in several variables. This can be seen as the ball-analogue of a formula originally invented by Richter. We capitalize on this formula to study Dirichlet-type spaces on the unit ball and joint $2$-isometries.
Let $g$ be a holomorphic function of the unit ball $B$ in the $n$-dimensional space, and denote by $T_g$ and $I_g$ the induced extended Ces$\acute{a}$ro operator and another integral operator. The boundedness and compactness of $T_g$ and $I_g$ acting on the Zygmund spaces in the unit ball are discussed and necessary and sufficient conditions are given in this paper.
In this paper we continue the study of free holomorphic functions on the unit ball of B(H)^n, where B(H) is the algebra of all bounded linear operators on a Hilbert space H. Several classical results from complex analysis have free analogues in our noncommutative setting.
Some new characterizations on Carleson measures for weighted Bergman spaces on the unit ball involving product of functions are obtained. For these we characterize bounded and compact Toeplitz operators between weighted Bergman spaces. The above results are applied to characterize bounded and compact extended Cesàro operators and pointwise multiplication operators. The results are new even in the case of the unit disk.
We extend results of Bénéteau, Condori, Liaw, Seco, and the author concerning cyclic vectors in Dirichlet-type spaces to the setting of the unit ball, identifying some classes of cyclic and non-cyclic functions, and noting the necessity of certain capacity conditions.
This paper focuses on the embedding problem of linear fractional maps which explains when a linear fractional self-map of $B_{N}$ can be a member of a semigroup of holomorphic self-maps on the unit ball $B_{N}$ of the complex $N$-dimensional Euclidean space $\mathbb{C}^{N}$.
The boundedness of $P_ω:L^\infty(\mathbb{B})\to \mathcal{B}(\mathbb{B})$ and $P_ω(P_ω^+):L^p(\mathbb{B},\upsilon dV)\to L^p(\mathbb{B},\upsilon dV)$ on the unit ball of $\mathbb{C}^n$ with $p>1$ and $ω,\upsilon\in \mathcal{D}$ are investigated in this paper.
We determine those norms on B(H) whose unit ball is C*-convex. We call them M-norms and show that the class of M-norms less than a given norm enjoys a maximum element. These minimum and maximum elements will be determined in some cases. Finally, we give a constructive result to obtain M-norms on B(H).
A \emph{multicontraction} on a Hilbert space $\HH$ is an $n$-tuple of operators $T=(T_1,...,T_n)$ acting on $\HH$, such that $\sum_{i=1}^n T_i T_i^*\le \1_\HH$. We obtain some results related to the characteristic function of a commuting multicontraction, most notably discussing its behaviour with respect to the action of the analytic automorphisms of the unit ball.
We obtain some estimates for norm and essential norm of the difference of two composition operators between weighted Bergman spaces $A^p_α$ and $A^q_β$ on the unit ball. In particular, we completely characterize the boundedness and compactness of $C_φ-C_ψ: A^p_α\to A^q_β$ for full range $0<p,q<\infty$, $-1<α,β<\infty$.
We adapt Sarason's proof of the Julia-Caratheodory theorem to the class of Schur-Agler mappings of the unit ball, obtaining a strengthened form of this theorem. In particular those quantities which appear in the classical theorem and depend only on the component of the mapping in the complex normal direction have K-limits (not just restricted K-limits) at the boundary.
Bergman-type integral operators are classical operators in complex analysis and operator theory. Recently, the first author and his collaborator \cite{DiW} completely characterized the $L^p$-$L^q$ boundedness of Bergman-type integral operators $K_α,K_α^+$ and the $L^p$-$L^q$ compactness of $K_α$ on the unit ball. In this paper, we will use a substantially new method to completely characterize the $L^p$-$L^q$ compactness of $K_α^+,$ but also prove that the $L^p$-$L^q$ compactness of operators $K_α,K_α^+$ is in fact equivalent. Moreover, we completely characterize Schatten class and Macaev class…
