The translation of an operator is defined by using conjugation with time-frequency shifts. Thus, one can define $Λ$-shift-invariant subspaces of Hilbert-Schmidt operators, finitely generated, with respect to a lattice $Λ$ in $\mathbb{R}^{2d}$. These spaces can be seen as a generalization of classical shift-invariant subspaces of square integrable functions. Obtaining sampling results for these subspaces appears as a natural question that can be motivated by the problem of channel estimation in wireless communications. These sampling results are obtained in the light of the frame theory in a separable Hilbert space.
The article is devoted to the problem of Hilbert-Schmidt type analytic extensions in Hardy spaces over the infinite-dimensional unitary matrix group endowed with an invariant probability measure. An orthogonal basis of Hilbert-Schmidt polynomials, reproducing kernels, integral formulas and boundary values are investigated.
In this paper we establish Parseval type identities and surprising new inequalities for Hilbert-Schmidt frames. Our results generalize and improve the remarkable results which have been obtained by Balan et al. and G{ă}vru{ţ}a.
The current work generalizes the author's previous work on the infinite-dimensional Alpha Log-Determinant (Log-Det) divergences and Alpha-Beta Log-Det divergences, defined on the set of positive definite unitized trace class operators on a Hilbert space, to the entire Hilbert manifold of positive definite unitized Hilbert-Schmidt operators. This generalization is carried out via the introduction of the extended Hilbert-Carleman determinant for unitized Hilbert-Schmidt operators, in addition to the previously introduced extended Fredholm determinant for unitized trace class operators. The resul…
In this work we discuss several ways to extend to the context of Banach spaces the notion of Hilbert-Schmidt operators: $p$-summing operators, $γ$-summing or $γ$-radonifying operators, weakly $*1$-nuclear operators and classes of operators defined via factorization properties. We introduce the class $PS_2(E; F)$ of pre-Hilbert-Schmidt operators as the class of all operators $u:E\to F$ such that $w\circ u \circ v$ is Hilbert-Schmidt for every bounded operator $v: H_1\to E$ and every bounded operator $w:F\to H_2$, where $H_1$ et $H_2$ are Hilbert spaces. Besides the trivial case where one of the…
Current work defines Schmidt representation of a bilinear operator $T: H_1 \times H_2 \rightarrow K$, where $H_1, H_2$ and $K$ are separable Hilbert spaces. Introducing the concept of singular value and ordered singular value, we prove that if $T$ is compact, and its singular values are ordered, then $T$ has a Schmidt representation on real Hilbert spaces. We prove that the hypothesis of existence of ordered singular values is fundamental.
The concept of translation of an operator allows to consider the analogous of shift-invariant subspaces in the class of Hilbert-Schmidt operators. Thus, we extend the concept of average sampling to this new setting, and we obtain the corresponding sampling formulas. The key point here is the use of the Weyl transform, a unitary mapping between the space of square integrable functions in the phase space $\mathbb{R}^d\times \widehat{\mathbb{R}}^d$ and the Hilbert space of Hilbert-Schmidt operators on $L^2(\mathbb{R}^d)$, which permits to take advantage of some well established sampling results.
In this paper an approximation of the image of the closed ball of the space $L_p$ $(p>1)$ centered at the origin with radius $r$ under Hilbert-Schmidt integral operator $F(\cdot):L_p\rightarrow L_q$ $\displaystyle \left(\frac{1}{p}+\frac{1}{q}=1\right)$ is presented. An error estimation for given approximation is obtained.
A closed subspace $\mathcal{M}$ of the Hardy space $H^2(\mathbb{D}^2)$ over the bidisk is called a submodule if it is invariant under multiplication by coordinate functions $z_1$ and $z_2$. Whether every finitely generated submodule is Hilbert-Schmidt is an unsolved problem. This paper proves that every finitely generated submodule $\mathcal{M}$ containing $z_1 - φ(z_2)$ is Hilbert-Schmidt, where $φ$ is any finite Blaschke product. Some other related topics such as fringe operator and Fredholm index are also discussed.
In this paper, we study the Hilbert$-$Schmidt frame (HS-frame) theory for separable Hilbert spaces. We first present some characterizations of HS-frames and prove that HS-frames share many important properties with frames. Then, we show how the inverse of the HS-frame operator can be approximated using finite-dimensional methods. Finally, we present a classical perturbation result and prove that HS-frames are stable under small perturbations.
We define angle $Θ_{X,Y}$ between non-zero Hilbert--Schmidt operators $X$ and $Y$ by $\cosΘ_{_{X,Y}} = \frac{{\rm Re}{\rm Tr}(Y^*X)}{{\|X\|}_{_2}{\|Y\|}_{_2}}$, and give some of its essentially properties. It is shown, among other things, that \begin{align*} \big|\cosΘ_{_{X,Y}}\big|\leq \min\left\{\sqrt{\cosΘ_{_{|X^*|,|Y^*|}}}, \sqrt{\cosΘ_{_{|X|,|Y|}}}\right\}. \end{align*} It enables us to provide alternative proof of some well-known inequalities for the Hilbert--Schmidt norm. In particular, we apply this inequality to prove Lee's conjecture [Linear Algebra Appl. 433 (2010), no.~3, 580--584]…
Let $A$ be a $m\times m$ complex matrix with zero trace. Then there are $m\times m$ matrices $B$ and $C$ such that $A=[B,C]$ and $\|B\|\|C\|_2\le (\log m+O(1))^{1/2}\|A\|_2$ where $\|D\|$ is the norm of $D$ as an operator on $\ell_2^m$ and $\|D\|_2$ is the Hilbert--Schmidt norm of $D$. Moreover, the matrix $B$ can be taken to be normal. Conversely there is a zero trace $m\times m$ matrix $A$ such that whenever $A=[B,C]$, $\|B\|\|C\|_2\ge |\log m-O(1)|^{1/2}\|A\|_2$ for some absolute constant $c>0$.
Let $\{λ_n\}_n \in \ell^\infty(\mathbb{N})$. In 1960, R. Schatten \cite{SCHATTEN} studied operators of the form $\sum_{n=1}^{\infty}λ_n (x_n\otimes \bar{y_n})$, where $\{x_n\}_n$, $\{y_n\}_n$ are orthonormal sequences in a Hilbert space. In 2007, P. Balazs \cite{BALAZS3} generalized this by replacing $\{x_n\}_n$ and $\{y_n\}_n$ by Bessel sequences. In this paper, we generalize this by studying the operators of the form $\sum_{n=1}^{\infty}λ_n (A^*_nx_n\otimes \bar{B^*_ny_n})$, where $\{A_n\}_n$ and $\{B_n\}_n$ are operator-valued Bessel sequences and $\{x_n\}_n$, $\{y_n\}_n$ are sequences in t…
We give a short proof of the main result of our previous paper [2]: every Schmidt subspace of a Hankel operator is the image of a model space by an isometric multiplier. This class of subspaces is closely related to nearly $S^*$-invariant subspaces, and our proof uses Hitt's theorem on the structure of such subspaces. We also give a formula for the action of a Hankel operator on its Schmidt subspace.
In this paper we introduce and study pseudo-differential operators with operator valued symbols on the abstract Heisenberg group $\mathbb{H}(G):=G \times \widehat{G} \times \mathbb{T},$ where $G$ a locally compact abelian group with its dual group $\widehat{G}$. We obtain a necessary and sufficient condition on symbols for which these operators are in the class of Hilbert-Schmidt operators. As a key step in proving this we derive a trace formula for the trace class $j$-Weyl transform, $j \in \mathbb{Z}^*$ with symbols in $L^{2}(G\times \widehat{G}).$ We go on to present a characterization of t…
This work presents a parametrized family of distances, namely the Alpha Procrustes distances, on the set of symmetric, positive definite (SPD) matrices. The Alpha Procrustes distances provide a unified formulation encompassing both the Bures-Wasserstein and Log-Euclidean distances between SPD matrices. We show that the Alpha Procrustes distances are the Riemannian distances corresponding to a family of Riemannian metrics on the manifold of SPD matrices, which encompass both the Log-Euclidean and Wasserstein Riemannian metrics. This formulation is then generalized to the set of positive definit…
In this paper we study the small-$λ$ spectral asymptotics of an integral operator $\mathscr{K}$ defined on two multi-intervals $J$ and $E$, when the multi-intervals touch each other (but their interiors are disjoint). The operator $\mathscr{K}$ is closely related to the multi-interval Finite Hilbert Transform (FHT). This case can be viewed as a singular limit of self-adjoint Hilbert-Schmidt integral operators with so-called integrable kernels, where the limiting operator is still bounded, but has a continuous spectral component. The regular case when $\text{dist}(J,E)>0$, and $\mathscr{K}$ is …
For a bounded Hankel matrix $Γ$, we describe the structure of the Schmidt subspaces of $Γ$, namely the eigenspaces of $Γ^* Γ$ corresponding to non zero eigenvalues. We prove that these subspaces are in correspondence with weighted model spaces in the Hardy space on the unit circle. Here we use the term "weighted model space" to describe the range of an isometric multiplier acting on a model space. Further, we obtain similar results for Hankel operators acting in the Hardy space on the real line. Finally, we give a streamlined proof of the Adamyan-Arov-Krein theorem using the language of weight…
Let $\mathbb{B}(\mathcal{H})$ be the algebra of all bounded linear operators on a Hilbert space $\mathcal{H}$ and let $N(\cdot)$ be a norm on $\mathbb{B}(\mathcal{H})$. For every $0\leq ν\leq 1$, we introduce the $w_{_{(N,ν)}}(A)$ as an extension of the classical numerical radius by \begin{align*} w_{_{(N,ν)}}(A):= \displaystyle{\sup_{θ\in \mathbb{R}}} N\left(νe^{iθ}A + (1-ν)e^{-iθ}A^*\right) \end{align*} and investigate basic properties of this notion and prove inequalities involving it. In particular, when $N(\cdot)$ is the Hilbert--Schmidt norm ${\|\!\cdot\!\|}_{2}$, we present several the …
We give necessary and sufficient conditions for the sum of n subspaces of a Hilbert space to be closed. We also present various properties of n-tuples of subspaces with closed sum.
By applying methods of Duhamel algebra and reproducing kernels, we prove that every linear bounded operator on the Hardy-Hilbert space H^{2}(D) has a nontrivial invariant subspace. This solves affirmatively the Invariant Subspace Problem in the Hilbert space.
We study finite systems of subspaces of a complex Hilbert space such that each pair of subspaces satisfies a certain condition as described in the following. For each subspace excepting the first one an angle between this subspace and the first one is fixed. The set of all subspaces excluding the first one is divided into disjoint subsets. Each such subset consists of one or two subspaces. Subspaces from different subsets are orthogonal and orthogonal projections onto subspaces from the same subset commute.
We characterize operators $T=PQ$ ($P,Q$ orthogonal projections in a Hilbert space $H$) which have a singular value decomposition. A spatial characterizations is given: this condition occurs if and only if there exist orthonormal bases $\{ψ_n\}$ of $R(P)$ and $\{ξ_n\}$ of $R(Q)$ such that $\langleξ_n,ψ_m\rangle=0$ if $n\ne m$. Also it is shown that this is equivalent to $A=P-Q$ being diagonalizable. Several examples are studied, relating Toeplitz, Hankel and Wiener-Hopf operators to this condition. We also examine the relationship with the differential geometry of the Grassmann manifold of unde…
We prove that for every semigroup of Schwarz maps on the von~Neumann algebra of all bounded linear operators on a Hilbert space which has a subinvariant faithful normal state there exists an associated semigroup of contractions on the space of Hilbert-Schmidt operators of the Hilbert space. Moreover, we show that if the original semigroup is weak$^*$ continuous then the associated semigroup is strongly continuous. We introduce the notion of the extended generator of a semigroup on the bounded operators of a Hilbert space with respect to an orthonormal basis of the Hilbert space. We describe th…
If matrices almost satisfying a group relation are close to matrices exactly satisfying the relation, then we say that a group is matricially stable. Here "almost" and "close" are in terms of the Hilbert-Schmidt norm. Using tracial 2-norm on $II_1$-factors we similarly define $II_1$-factor stability for groups. Our main result is that all 1-relator groups with non-trivial center are $II_{1}$-factor stable. Many of them are also matricially stable and RFD. For amenable groups we give a complete characterization of matricial stability in terms of the following approximation property for characte…
We present a general construction of non-Gaussian probability measures on the space of distributional kernels obeying a natural extension of the Osterwalder-Schrader axioms of Euclidean quantum field theory in arbitrary space-time dimension $d$. These measures may be interpreted as corresponding to scalar massive quantum fields with polynomial self-interaction. As a consequence, we obtain examples of non-free models satisfying the Haag-Kastler axioms of algebraic quantum field theory for arbitrary $d$. When $d<4$ we are able to transfer the measures to the space of distributions and verify the…
The 'relativistic' Heun equation is an 8-coupling difference equation that generalizes the 4-coupling Heun differential equation. It can be viewed as the time-independent Schrödinger equation for an analytic difference operator introduced by van Diejen. We study Hilbert space features of this operator and its 'modular partner', based on an in-depth analysis of the eigenvectors of a Hilbert-Schmidt integral operator whose integral kernel has a previously known relation to the two difference operators. With suitable restrictions on the parameters, we show that the commuting difference operators …
In the setting of operators on Hilbert spaces, we prove that every quasinilpotent operator has a non-trivial closed invariant subspace if and only if every pair of idempotents with a quasinilpotent commutator has a non-trivial common closed invariant subspace. We also present a geometric characterization of invariant subspaces of idempotents and classify operators that are essentially idempotent.
We study the relative position of four subspaces in a Hilbert space. For any positive integer n, we give an example of exotic indecomposable system of four subspaces in a Hilbert space whose defect is (2n+1)/3. By an exotic system, we mean a system which is not isomorphic to any closed operator system under any permutation of subspaces. We construct the examples by a help of certain nice sequences used by Jiang and Wang in their study of strongly irreducible operators.
We study the relative position of three subspaces in a separable infinite-dimensional Hilbert space. In the finite-dimensional case, Brenner described the general position of three subspaces completely. We extend it to a certain class of three subspaces in an infinite-dimensional Hilbert space. We also give a partial result which gives a condition on a system to have a (dense) decomposition containing a pentagon.
For a shift operator $T$ with finite multiplicity acting on a separable infinite dimensional Hilbert space we represent its nearly $T^{-1}$ invariant subspaces in terms of invariant subspaces under the backward shift. Going further, given any finite Blaschke product $B$, we give a description of the nearly $T_{B}^{-1}$ invariant subspaces for the operator $T_B$ of multiplication by $B$ in a scale of Dirichlet-type spaces.
We prove that any product of two non-abelian free groups, $Γ=\mathbb F_m\times\mathbb F_k$, for $m,k\geq 2$, is not Hilbert-Schmidt stable. This means that there exist asymptotic representations $π_n:Γ\rightarrow \text{U}({d_n})$ with respect to the normalized Hilbert-Schmidt norm which are not close to actual representations. As a consequence, we prove the existence of contraction matrices $A,B$ such that $A$ almost commutes with $B$ and $B^*$, with respect to the normalized Hilbert-Schmidt norm, but $A,B$ are not close to any matrices $A',B'$ such that $A'$ commutes with $B'$ and $B'^*$. Thi…
Let T be a C_{\cdot 0}-contraction on a Hilbert space H and S be a non-trivial closed subspace of H. We prove that S is a T-invariant subspace of H if and only if there exists a Hilbert space D and a partially isometric operator Π: H^2_D(\mathbb{D}) \raro H such that ΠM_z = T Πand that S = ran Π, or equivalently, P_S = ΠΠ^*. As an application we completely classify the shift-invariant subspaces of C_{\cdot 0}-contractive and analytic reproducing kernel Hilbert spaces over the unit disc. Our results also includes the case of weighted Bergman spaces over the unit disk.
