Hilbert-Schmidt Operators Research Articles

Composition operators between different Dirichlet spaces

In this article, we study the boundedness of composition operators between strong and weak version vector-valued weighted Dirichlet spaces. Some sufficient and necessary conditions for such composition operators to be bounded are obtained exactly, which are different from the scalar-valued case and agree with the characterizations for these composition operators to be Hilbert–Schmidt operators between the corresponding scalar-valued Dirichlet spaces. As a by-product, we show indirectly that these strong and weak version Dirichlet spaces are different significatively, and obtain the equivalence of some analytic properties of an analytic self-map of the unit disk.

Hilbert-Schmidt Hankel Operators and Berezin Iteration

Let $H$ be a reproducing kernel Hilbert space contained in a wider space $L^2(X,\mu)$. We study the Hilbert-Schmidt property of Hankel operators $H_g$ on $H$ with bounded symbol $g$ by analyzing the behavior of the iterated Berezin transform. We determine symbol classes $\mathcal{S}$ such that for $g\in \mathcal{S}$ the Hilbert-Schmidt property of $H_g$ implies that $H_{\bar{g}}$ is a Hilbert-Schmidt operator as well and there is a norm estimate of the form $\|H_{\bar{g}}\|_{\text{HS}}\leq C\cdot \| H_g\|_{\text{HS}}$. Finally, applications to the case of Bergman spaces over strictly pseudo convex domains in $\mathbb{C}^n$, the Fock space, the pluri-harmonic Fock space and spaces of holomorphic functions on a quadric are given.

Hopf–Rinow theorem in the Sato Grassmannian

Let U 2 ( H ) be the Banach–Lie group of unitary operators in the Hilbert space H which are Hilbert–Schmidt perturbations of the identity 1. In this paper we study the geometry of the unitary orbit { u p u ∗ : u ∈ U 2 ( H ) } , of an infinite projection p in H . This orbit coincides with the connected component of p in the Hilbert–Schmidt restricted Grassmannian Gr res ( p ) (also known in the literature as the Sato Grassmannian) corresponding to the polarization H = p ( H ) ⊕ p ( H ) ⊥ . It is known that the components of Gr res ( p ) are differentiable manifolds. Here we give a simple proof of the fact that Gr res 0 ( p ) is a smooth submanifold of the affine Hilbert space p + B 2 ( H ) , where B 2 ( H ) denotes the space of Hilbert–Schmidt operators of H . Also we show that Gr res 0 ( p ) is a homogeneous reductive space. We introduce a natural metric, which consists in endowing every tangent space with the trace inner product, and consider its Levi-Civita connection. This connection has been considered before, for instance its sectional curvature has been computed. We show that the Levi-Civita connection coincides with a linear connection induced by the reductive structure, a fact which allows for the easy computation of the geodesic curves. We prove that the geodesics of the connection, which are of the form γ ( t ) = e t z p e − t z , for z a p-co-diagonal anti-hermitic element of B 2 ( H ) , have minimal length provided that ‖ z ‖ ⩽ π / 2 . Note that the condition is given in terms of the usual operator norm, a fact which implies that there exist minimal geodesics of arbitrary length. Also we show that any two points p 1 , p 2 ∈ Gr res 0 ( p ) are joined by a minimal geodesic. If moreover ‖ p 1 − p 2 ‖ < 1 , the minimal geodesic is unique. Finally, we replace the 2-norm by the k-Schatten norm ( k > 2 ), and prove that the geodesics are also minimal for these norms, up to a critical value of t, which is estimated also in terms of the usual operator norm. In the process, minimality results in the k-norms are also obtained for the group U 2 ( H ) .

Fréchet-differentiation of functions of operators with application to testing the equality of two covariance operators

It is well-known that the sample covariance operator converges in distribution in the Hilbert space of Hilbert-Schmidt operators, and that this result entails the asymptotic distribution of simple eigenvalues and corresponding eigenvectors. Several estimators and test statistics for the analysis of functional data require the asymptotic distribution of eigenvalues and eigenvectors of certain functions of sample covariance operators. To obtain such a result, it turns out that the asymptotic distribution of such a function of the sample covariance operator is a prerequisite. In this paper we briefly review the Fréchet-derivative of functions of operators and an ensuing delta-method to solve this problem. The results are applied to obtain the asymptotic distribution of a statistic for testing the equality of two covariance operators.

A Note on Subnormal and Hyponormal Derivations

In this note we prove that if A and B are subnormal operators and X is a bounded linear operator such that AX XB is a Hilbert-Schmidt operator, then f(A)X Xf(B) is also a Hilbert-Schmidt operator and ||f(A)X Xf(B)||2 L||AX XB||2, for f belonging to a certain class of functions. Furthermore, we investigate the similar problem in the case that S, T are hyponormal operators and X 2 L(H) is such that SX XT belongs to a norm ideal (J, || · ||J) and prove that f(S)X Xf(T) 2 J and ||f(S)X Xf(T)||J C ||SX XT||J, for f in a certain class of functions.

Frame transforms, star products and quantum mechanics on phase space

Using the notions of frame transform and of square integrable projective representation of a locally compact group G, we introduce a class of isometries (tight frame transforms) from the space of Hilbert–Schmidt operators in the carrier Hilbert space of the representation into the space of square integrable functions on the direct product group G × G. These transforms have remarkable properties. In particular, their ranges are reproducing kernel Hilbert spaces endowed with a suitable ‘star product’ which mimics, at the level of functions, the original product of operators. A ‘phase space formulation’ of quantum mechanics relying on the frame transforms introduced in the present paper, and the link of these maps with both the Wigner transform and the wavelet transform are discussed.

HILBERT–SCHMIDT OPERATORS AND FRAMES — CLASSIFICATION, BEST APPROXIMATION BY MULTIPLIERS AND ALGORITHMS

In this paper we deal with the theory of Hilbert–Schmidt operators, when the usual choice of orthonormal basis, on the associated Hilbert spaces, is replaced by frames. We More precisely, we provide a necessary and sufficient condition for an operator to be Hilbert–Schmidt, based on its action on the elements of a frame (i.e. an operator T is [Formula: see text] if and only if the sum of the squared norms of T applied on the elements of the frame is finite). Also, we construct Bessel sequences, frames and Riesz bases of [Formula: see text] operators using tensor products of the same sequences in the associated Hilbert spaces. We state how the [Formula: see text] inner product of an arbitrary operator and a rank one operator can be calculated in an efficient way; and we use this result to provide a numerically efficient algorithm to find the best approximation, in the Hilbert–Schmidt sense, of an arbitrary matrix, by a so-called frame multiplier (i.e. an operator which act diagonally on the frame analysis coefficients). Finally, we give some simple examples using Gabor and wavelet frames, introducing in this way wavelet multipliers.

Some Estimates of Certain Subnormal and Hyponormal Derivations

We prove that if and are subnormal operators and is a bounded linear operator such that is a Hilbert-Schmidt operator, then is also a Hilbert-Schmidt operator and for belongs to a certain class of functions. Furthermore, we investigate the similar problem in the case that , are hyponormal operators and is such that belongs to a norm ideal , and we prove that and for being in a certain class of functions.

On the Relation between States and Maps in Infinite Dimensions

Relations between states and maps, which are known for quantum systems in finite-dimensional Hilbert spaces, are formulated rigorously in geometrical terms with no use of coordinate (matrix) interpretation. In a tensor product realization they are represented simply by a permutation of factors. This leads to natural generalizations for infinite-dimensional Hilbert spaces and a simple proof of a generalized Choi Theorem. The natural framework is based on spaces of Hilbert-Schmidt operators [Formula: see text] and the corresponding tensor products [Formula: see text] of Hilbert spaces. It is proved that the corresponding isomorphisms cannot be naturally extended to compact (or bounded) operators, nor reduced to the trace-class operators. On the other hand, it is proven that there is a natural continuous map [Formula: see text] from trace-class operators on [Formula: see text] (with the nuclear norm) into compact operators mapping the space of all bounded operators on [Formula: see text] into trace class operators on [Formula: see text] (with the operator-norm). Also in the infinite-dimensional context, the Schmidt measure of entanglement and multipartite generalizations of state-maps relations are considered in the paper.

CONTINUITY WITH RESPECT TO SYMBOLS OF COMPOSITION OPERATORS ON THE WEIGHTED BERGMAN SPACE

Let $\alpha \gt -1$, $U$ be the open unit disk in $\mathbb C$ and denote by $H(U)$ the set of all holomorphic functions on $U$. Let $C_\varphi$ be a composition operator induced by an analytic self-map $\varphi$ of $U$. Composition operators $C_\varphi$ on the weighted Hilbert Bergman space ${\mathcal A}^2_\alpha(U) = \big\{f \in H(U) \;|\; \int_U |f(z)|^2(1-|z|^2)^\alpha dm(z) \lt \infty \big\}$ are considered. We investigate when convergence of sequences $(\varphi_n)$ of symbols to a given symbol $\varphi$, implies the convergence of the induced composition operators. We give a necessary and sufficient condition for a sequence of Hilbert-Schmidt composition operators $(C_{\varphi_n})$ to converge in Hilbert-Schmidt norm to $C_\varphi$, and we obtain a sufficient condition for convergence in operator norm.

A perturbation problem for the shift semigroup

Motivated by B. Tsirelson's construction of E 0 -semigroups of type III, we investigate a C 0 -semigroup acting on the space of square integrable functions of the half line whose difference from the shift semigroup is a Hilbert–Schmidt operator. We give a description of such semigroups in terms of analytic functions on the right-half plane and construct several examples.

A discrete model for the efficient analysis of time-varying narrowband communication channels

We derive an efficient numerical algorithm for the analysis of certain classes of Hilbert–Schmidt operators that naturally occur in models of wireless radio and sonar communications channels. We show that many narrowband finite lifelength systems such as wireless radio communications can be well modelled by smooth and compactly supported spreading functions. Further, we exploit this fact to derive a fast algorithm for computing the matrix representation of such operators with respect to well time-frequency localized Gabor bases (such as pulseshaped OFDM bases). Hereby we use a minimum of approximations, simplifications, and assumptions on the channel. Moreover, we use a multivariate setting to allow for applications to, for example, antenna arrays. The derived algorithm and software can be used, for example, for comparing how different system settings and pulse shapes affect the diagonalization properties of an OFDM system acting on a given channel.

Nonpositive curvature: A geometrical approach to Hilbert–Schmidt operators

We give a Riemannian structure to the set Σ of positive invertible unitized Hilbert–Schmidt operators, by means of the trace inner product. This metric makes of Σ a nonpositively curved, simply connected and metrically complete Hilbert manifold. The manifold Σ is a universal model for symmetric spaces of the noncompact type: any such space can be isometrically embedded into Σ. We give an intrinsic algebraic characterization of convex closed submanifolds M. We study the group of isometries of such submanifolds: we prove that G M , the Banach–Lie group generated by M, acts isometrically and transitively on M. Moreover, G M admits a polar decomposition relative to M, namely G M ≃ M × K as Hilbert manifolds (here K is the isotropy of p = 1 for the action I g : p ↦ g p g ∗ ), and also G M / K ≃ M so M is an homogeneous space. We obtain several decomposition theorems by means of geodesically convex submanifolds M. These decompositions are obtained via a nonlinear but analytic orthogonal projection Π M : Σ → M , a map which is a contraction for the geodesic distance. As a byproduct, we prove the isomorphism N M ≃ Σ (here NM stands for the normal bundle of a convex closed submanifold M). Writing down the factorizations for fixed e a , we obtain e a = e x e v e x with e x ∈ M and v orthogonal to M at p = 1 . As a corollary we obtain decompositions for the full group of invertible elements G ≃ M × exp ( T 1 M ⊥ ) × K .

A central limit type theorem for a class of particle filters

The optimal fllter … = f…t; t ‚ 0g for a general observation model is approximated by a probability measure valued process …n = f…n ; t ‚ 0g. The process … n is the empirical measure of a system of weighted particles that at time 0 consists of n particles. The particles branch at equally spaced time instances jn i2fi where j = 1; 2; ::: and 0 < fi < 1. We prove the convergence of the process … n to … and derive sharp upper bounds for the mean square error. We also prove a central limit theorem to characterize the convergence rate of the approximate fllter. A similar result is obtained for the unweighted, unnormalized version introduced in (8). As a corollary, we show that fi = 1 is the optimal exponent for that version. The approximation of the optimal nonlinear fllter by means of particle approx- imations has been studied extensively in last ten years (see, for example, (5), (7), (10), (12) and the references therein). The use of particle approximations stems from the fact that the unnormalized fllter can be approximated by a weighted particle system. Since the weights have variances which grow exponentially fast, the particle system needs to be corrected after small time steps to control the error. At each time step, the particles will be replaced by a random number of \ofisprings. The expected number of ofisprings is the weight of the corresponding particle decided according to its path during the period prior to that time step. In the following, we will work within a very general flltering framework. Namely, we will assume that the observation process takes values in a space of measures (rather than the usual k-dimensional Euclidean space). In particular, the observa- tion process can be given by a random measure in space and time. Moreover, we will allow the observation and signal noises to be correlated. Let us now introduce the flltering model in more detail. Let (›;F; (Ft) t‚0 ;P0) be a flltered probability space and (H; h¢;¢i H ), respec- tively i H1; h¢;¢i H1 ¢ be two real separable Hilbert spaces such that H µ H1 and the injection from H to H1 is a Hilbert-Schmidt operator. On (›;F; (Ft) t‚0 ;P0) we deflne an Ft-adaptedd-dimensional Brownian motion B = fBt;Ft; t ‚ 0g and an Ft-adapted H-cylindrical Brownian motion (H-c.B.m.) W = fWt;Ft; t ‚ 0g

Learning Theory Estimates via Integral Operators and Their Approximations

The regression problem in learning theory is investigated with least square Tikhonov regularization schemes in reproducing kernel Hilbert spaces (RKHS). We follow our previous work and apply the sampling operator to the error analysis in both the RKHS norm and the L2 norm. The tool for estimating the sample error is a Bennet inequality for random variables with values in Hilbert spaces. By taking the Hilbert space to be the one consisting of Hilbert-Schmidt operators in the RKHS, we improve the error bounds in the L2 metric, motivated by an idea of Caponnetto and de Vito. The error bounds we derive in the RKHS norm, together with a Tsybakov function we discuss here, yield interesting applications to the error analysis of the (binary) classification problem, since the RKHS metric controls the one for the uniform convergence.

Study of the initial value problems appearing in a method of factorization of second-order elliptic boundary value problems

In [J. Henry, A.M. Ramos, Factorization of second order elliptic boundary value problems by dynamic programming, Nonlinear Analysis. Theory, Methods & Applications 59 (2004) 629–647] we presented a method for factorizing a second-order boundary value problem into a system of uncoupled first-order initial value problems, together with a nonlinear Riccati type equation for functional operators. A weak sense was given to that system but we did not perform a direct study of those equations. This factorization utilizes either the Neumann to Dirichlet (NtD) operator or the Dirichlet to Neumann (DtN) operator, which satisfy a Riccati equation. Here we consider the framework of Hilbert–Schmidt operators, which provides tools for a direct study of this Riccati type equation. Once we have solved the system of Cauchy problems, we show that its solution solves the original second-order boundary value problem. Finally, we indicate how this techniques can be used to find suitable transparent conditions.

Supercyclicity in the operator algebra using Hilbert-Schmidt operators

We prove that the Supercylicity Criterion for any operatorT on a Hilbert space is equivalent to the supercyclicity of the left multiplication operator induced byT in the strong operator topology.

CONTINUITY WITH RESPECT TO SYMBOLS OF COMPOSITION OPERATORS ON THE WEIGHTED BERGMAN SPACE

Let $\alpha>-1,$ $U$ be the open unit disk in $\mathbb C$ and denote by $H(U)$ the set of all holomorphic functions on $U.$ Let $C_\varphi$ be a composition operator induced by an analytic self-map $\varphi$ of $U.$ Composition operators $C_\varphi$ on the weighted Hilbert Bergman space ${\mathcal A}^2_\alpha(U)=\big\{f\in H(U)\;|\; \int_U |f(z)|^2(1-|z|^2)^\alpha dm(z)<\infty\big\}$ are considered. We investigate when convergence of sequences $(\varphi_n)$ of symbols to a given symbol $\varphi,$ implies the convergence of the induced composition operators. We give a necessary and sufficient condition for a sequence of Hilbert-Schmidt composition operators $(C_{\varphi_n})$ to converge in Hilbert-Schmidt norm to $C_\varphi,$ and we obtain a sufficient condition for convergence in operator norm.

Concentration of mass on the Schatten classes

Let 1 ⩽ p ⩽ ∞ and B ( S p n ) ˜ be the unit ball of the Schatten trace class of matrices on C n or on R n , normalized to have Lebesgue measure equal to one. We prove that λ ( { T ∈ B ( S p n ) ˜ : ‖ T ‖ HS n ⩾ c 1 t } ) ⩽ exp ( − c 2 t n k p ) for every t ⩾ 1 , where k p = min { 2 , 1 + p / 2 } , c 1 , c 2 > 1 are universal constants and λ is the Lebesgue measure. This concentration of mass inside a ball of radius proportional to n follows from an almost constant behaviour of the L q norms (with respect to the Lebesgue measure on B ( S p n ) ˜ ) of the Hilbert–Schmidt operator norm of T. The same concentration result holds for every classical ensembles of matrices like real symmetric matrices, Hermitian matrices, symplectic matrices or antisymmetric Hermitian matrices. The result is sharp when p = 1 and p ⩾ 2 .

Note on Hilbert-Schmidt composition operators on weighted Hardy spaces

We show that if Cφ is a Hilbert-Schmidt composition operator on an appropriately weighted Hardy space, then there exists a capacity, associated to the weight sequence of the space, so that the set on which the radial limit of φ is unimodular has capacity zero. This extends recent results by Gallardo-Gutierrez and Gonzalez. Let D be the open unit disk in the complex plane and suppose that (X, ‖ · ‖) is a Hilbert space of analytic functions on D. We say that X is a weighted Hardy space if the set {z : j = 0, 1, 2, . . . } of monomials is a complete orthogonal system. We put βj = ‖z‖. Then β := {βj} is called the weight sequence and X is denoted by H(β). Many classical function spaces are weighted Hardy spaces. For example, the standard Hardy spaceH, the α-Dirichlet space Dα, 0 ≤ α < 1, of all analytic functions whose first derivative is square integrable with respect to the measure (1 − |z|)dA(z), and the Bergman space A of all square integrable analytic functions are particular instances of H(β) with βj ≡ 1, βj ∼ (1 + j)(1−α)/2 and βj = (1 + j)−1/2 respectively. Now suppose that φ : D → D is an analytic self-map of the unit disk and consider the corresponding composition operator Cφ acting on H(β), i.e. Cφ(f) = f ◦ φ, f ∈ H(β). We are interested in the behavior of those φ which induce HilbertSchmidt composition operators. 2000 Mathematics Subject Classification. 47B33, 30C85, 31A20.