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We use the Maurey–Rosenthal factorization theorem to give the necessary and sufficient conditions that an operator on the Cartezian product of $ l_{1}\left (H\right) $ l 1 (H) -spaces with values in a Hilbert space to be multiple 2-summing. Various applications of this characterization are given. [ABSTRACT FROM AUTHOR]

Numerous works have been dedicated to the topic of m -isometries, including [2–6,14,18–20,27,47,48]. In this article, we introduce the concept of (m , N A) -isometry, where A is a non-zero operator and m is a positive integer, as an extension of the m -isometry class created by J. Alger and M. Stankus in the 1980s. We present some algebraic and spectral characteristics of (m , N A) -isometries. Additionally, we investigate the product of an (m , N A) -isometry by an (n , N B) -isometry, which enhances and broadens the previous work of Gu et al. on m -isometries [40]. Finally, we apply our main findings to elementary operators defined on the Hilbert-Schmidt class, which can be identified with a tensor product. This provides a new, less complicated, and non-combinatorial proof of Theorem 2.10 of [30]. [ABSTRACT FROM AUTHOR]

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In this paper we present necessary and sufficient conditions characterizing compact sets in the spaces of Nuclear and Hilbert-Schmidt operators on Hilbert spaces. Based on these results, we also characterize compact sets in the space of probability measures on separable Hilbert spaces. These results are then used in the study of optimal feedback control theory for stochastic systems. We prove existence of optimal feedback control laws for standard and several nonstandard control problems. Further, we present necessary conditions of optimality and an algorithm and related convergence theorem whereby optimal control laws can be constructed. We present also a result characterizing compact sets in the space of probability measures on separable Hilbert spaces. These results have interesting applications in control theory on infinite dimensional Banach spaces as presented in the last section. [ABSTRACT FROM AUTHOR]

In this paper, we study some operator theoretical properties of pseudo-differential operators with operator-valued symbols on the Heisenberg motion group. Specifically, we investigate L 2 – L p boundedness of pseudo-differential operators on the Heisenberg motion group for the range 2 ≤ p ≤ ∞. We also provide a necessary and sufficient condition on the operator-valued symbols in terms of λ-Weyl transforms such that the corresponding pseudo-differential operators on the Heisenberg motion group are in the class of Hilbert–Schmidt operators. As a consequence, we obtain a characterization of the trace class pseudo-differential operators on the Heisenberg motion group and provide a trace formula for these trace class operators. [ABSTRACT FROM AUTHOR]

Let B(H ) denote the C∗ -algebra of all bounded linear operators acting on a complex Hilbert space H. For A,B ∈ B(H ), define the bimultiplication operator M2,A,B on the class of Hilbert-Schmidt operators by M2,A,B(X) = AXB. It is known that if either A or B is hyponormal, then /W(M2,A,B) = /co W(A)W(B), where the bar and co stand for the closure and the convex hull, respectively and W(·) denotes the numerical range. In this paper, we give some conditions satisfied by A and B to have the following equality W0(M2,A,B) = co W0(A)W0(B), where W0(·) denotes the maximal numerical range. [ABSTRACT FROM AUTHOR]

Regularity estimates for an integral operator with a symmetric continuous kernel on a convex bounded domain are derived. The covariance of a mean-square continuous random field on the domain is an example of such an operator. The estimates are of the form of Hilbert–Schmidt norms of the integral operator and its square root, composed with fractional powers of an elliptic operator equipped with homogeneous boundary conditions of either Dirichlet or Neumann type. These types of estimates, which couple the regularity of the driving noise with the properties of the differential operator, have important implications for stochastic partial differential equations on bounded domains as well as their numerical approximations. The main tools used to derive the estimates are properties of reproducing kernel Hilbert spaces of functions on bounded domains along with Hilbert–Schmidt embeddings of Sobolev spaces. Both non-homogeneous and homogeneous kernels are considered. In the latter case, results in a general Schatten class norm are also provided. Important examples of homogeneous kernels covered by the results of the paper include the class of Matérn kernels. [ABSTRACT FROM AUTHOR]

Given input–output pairs of an elliptic partial differential equation (PDE) in three dimensions, we derive the first theoretically rigorous scheme for learning the associated Green's function G. By exploiting the hierarchical low-rank structure of G, we show that one can construct an approximant to G that converges almost surely and achieves a relative error of O (Γ ϵ - 1 / 2 log 3 (1 / ϵ) ϵ) using at most O (ϵ - 6 log 4 (1 / ϵ)) input–output training pairs with high probability, for any 0 < ϵ < 1 . The quantity 0 < Γ ϵ ≤ 1 characterizes the quality of the training dataset. Along the way, we extend the randomized singular value decomposition algorithm for learning matrices to Hilbert–Schmidt operators and characterize the quality of covariance kernels for PDE learning. [ABSTRACT FROM AUTHOR]

We study the composition operators on certain weighted Hilbert spaces H ω , consisting of analytic functions defined on D , that can be described as a weighted Dirichlet space with weight ω having certain properties. We will use the notion of vanishing ω -Carleson measure to characterize the compact composition operators on H ω . Also, we give some upper and lower estimates for the essential norm of these operators and investigate the Hilbert-Schmidt composition operators. [ABSTRACT FROM AUTHOR]

Abstract In 1990, Romero presented a beautiful formula for the projection onto the set of rectangular matrices with prescribed row and column sums. Variants of Romero’s formula were rediscovered by Khoury and by Glunt, Hayden, and Reams for bistochastic (square) matrices in 1998. These results have found various generalizations and applications. In this paper, we provide a formula for the more general problem of finding the projection onto the set of rectangular matrices with prescribed scaled row and column sums. Our approach is based on computing the Moore–Penrose inverse of a certain linear operator associated with the problem. In fact, our analysis holds even for Hilbert–Schmidt operators, and we do not have to assume consistency. We also perform numerical experiments featuring the new projection operator.

Let u and $\varphi $ be two analytic functions on the unit disk D such that $\varphi (D) \subset D$. A weighted composition operator $uC_{\varphi }$ induced by u and $\varphi $ is defined on $A^2_{\alpha }$ , the weighted Bergman space of D, by $uC_{\varphi }f := u \cdot f \circ \varphi $ for every $f \in A^2_{\alpha }$. We obtain sufficient conditions for the compactness of $uC_{\varphi }$ in terms of function-theoretic properties of u and $\varphi $. We also characterize when $uC_{\varphi }$ on $A^2_{\alpha }$ is Hilbert–Schmidt. In particular, the characterization is independent of $\alpha $ when $\varphi $ is an automorphism of D. Furthermore, we investigate the Hilbert–Schmidt difference of two weighted composition operators on $A^2_{\alpha }$. [ABSTRACT FROM AUTHOR]

M. Putinar proved that hyponormal operators are sub-scalar. In this note we will use Putinar's construction to prove that the generalized commutator T ∗ X ^ ^ - X ^ ^ S ∗ belongs to the Hilbert-Schmidt class whenever T X - X S does so with T , S ∗ hyponormal operators and X ^ ^ is some transform of X that depends on T and S. Also, we prove that the commutator T ∗ X ^ ^ - X ^ ^ T ∗ is Hilbert-Schmidt when T is a hyponormal operators of finite cyclic multiplicity and T X - X T is a Hilbert-Schmidt operator. [ABSTRACT FROM AUTHOR]

Let \(u\) and \(\varphi\) be two analytic functions on the unit disk \(\mathbb{D}\) such that \(\varphi(\mathbb{D}) \subset \mathbb{D}\). A weighted composition operator \(uC_{\varphi}\) induced by \(u\) and \(\varphi\) is defined on \(H^2\), the Hardy space of \(\mathbb{D}\), by \(uC_{\varphi}f := u \cdot f \circ \varphi\) for every \(f\) in \(H^2\). We obtain sufficient conditions for Hilbert-Schmidtness of \(uC_{\varphi}\) on \(H^2\) in terms of function-theoretic properties of \(u\) and \(\varphi\). Moreover, we characterize Hilbert-Schmidt difference of two weighted composition operators on \(H^2\).

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 R2d. 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. [ABSTRACT FROM AUTHOR]

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 R2d. 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. [ABSTRACT FROM AUTHOR]

Strassen's theorem circa 1965 gives necessary and sufficient conditions on the existence of a probability measure on two product spaces with given support and two marginals. In the case where each product space is finite, Strassen's theorem is reduced to a linear programming problem which can be solved using flow theory. A density matrix of bipartite quantum system is a quantum analog of a probability matrix on two finite product spaces. Partial traces of the density matrix are analogs of marginals. The support of the density matrix is its range. The analog of Strassen's theorem in this case can be stated and solved using semidefinite programming. The aim of this paper is to give analogs of Strassen's theorem to density trace class operators on a product of two separable Hilbert spaces, where at least one of the Hilbert spaces is infinite-dimensional. [ABSTRACT FROM AUTHOR]

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 resulting parametrized family of Alpha-Beta Log-Det divergences is general and contains many divergences between positive definite unitized Hilbert–Schmidt operators as special cases, including the infinite-dimensional affine-invariant Riemannian distance and the infinite-dimensional generalization of the symmetric Stein divergence. [ABSTRACT FROM AUTHOR]

Some trace inequalities of Cassels type for operators in Hilbert spaces are provided. Applications in connection to Grüss inequality and for convex functions of selfadjoint operators are also given.

Hilbert-Schmidt Independence Criterion (HSIC) measures statistical independence between two random variables. However, instead of measuring the statistical independence between two random variables directly, HSIC first transforms two random variables into two Reproducing Kernel Hilbert Spaces (RKHS) respectively and then measures the kernelled random variables by using Hilbert-Schmidt (HS) operators between the two RKHS. Since HSIC was first proposed around 2005, HSIC has found wide applications in machine learning. In this paper, a HSIC regularized Local Tangent Space Alignment algorithm (HSIC-LTSA) is proposed. LTSA is a well-known dimensionality reduction algorithm for local homeomorphism preservation. In HSIC-LTSA, behind the objective function of LTSA, HSIC between high-dimensional and dimension-reduced data is added as a regularization term. The proposed HSIC-LTSA has two contributions. First, HSIC-LTSA implements local homeomorphism preservation and global statistical correlation during dimensionality reduction. Secondly, HSIC-LTSA proposes a new way to apply HSIC: HSIC is used as a regularization term to be added to other machine learning algorithms. The experimental results presented in this paper show that HSIC-LTSA can achieve better performance than the original LTSA. [ABSTRACT FROM AUTHOR]

Let G be a compact Hausdorff group and let H be a closed subgroup of G. We introduce pseudo-differential operators with symbols on the homogeneous space G / H {G/H}. We present a necessary and sufficient condition on symbols for which these operators are in the class of Hilbert–Schmidt operators. We also give a characterization of and a trace formula for the trace class pseudo-differential operators on the homogeneous space G / H {G/H}. [ABSTRACT FROM AUTHOR]

Some trace inequalities of Shisha-Mond type for operators in Hilbert spaces are provided. Applications in connection to Grüss inequality and for convex functions of selfadjoint operators are also given.

We study the reproducing kernel Hilbert space with kernel kd , where d is a positive integer and k is the reproducing kernel of the analytic Dirichlet space.

Abstract In this paper we consider a special class of continuous bilinear operators acting in a product of Banach algebras of integrable functions with convolution product. In the literature, these bilinear operators are called ‘zero product preserving’, and they may be considered as a generalization of Lamperti operators. We prove a factorization theorem for this class, which establishes that each zero product preserving bilinear operator factors through a subalgebra of absolutely integrable functions. We obtain also compactness and summability properties for these operators under the assumption of some classical properties for the range spaces, as the Dunford–Pettis property or the Schur property and we give integral representations by some concavity properties of operators. Finally, we give some applications for integral transforms, and an integral representation for Hilbert–Schmidt operators. [ABSTRACT FROM AUTHOR]

The ability to finely manipulate a population of structurally identical dynamical systems is important for various emerging applications across a broad spectrum of fields in science and engineering. Robust control of such an ensemble is fundamentally challenging, because individual systems in the ensemble obey the same dynamical law but have different system parameter values, and only a broadcast control can be applied to the entire ensemble. In this paper, we establish controllability conditions for the finite-dimensional time-varying linear ensemble system in a Hilbert space, whose parameters are in an unbounded set. This work extends our previous study in one-parameter families of linear ensemble systems with the parameter lying in a one-dimensional compact set. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the subspace transitivity and subspace supercyclicity of tuples of left multiplication operators in the strong operator topology and in the norm of Hilbert-Schmidt operators. [ABSTRACT FROM AUTHOR]

Some Jensen's type trace inequalities for convex functions of selfadjoint operators in Hilbert spaces are provided. Applications for some convex functions of interest are also given. [ABSTRACT FROM AUTHOR]

Let G be the Green's function for the Airy operator We show that the integral operator defined by G is Hilbert-Schmidt and that the 2-modified Fredholm determinant [ABSTRACT FROM AUTHOR]

Let (Ω,Σ, µ) be a probability space. We give a few results about operators onL1(µ). Among these, we show that ifTis a bounded linear operator onL1(µ) which acts as a Hilbert-Schmidt operator onL2(µ), thenT:L1(µ)→ L1(µ) is representable. [ABSTRACT FROM AUTHOR]

In this paper we show that the property of disjoint supercyclic operators T, ⋯, T satisfying d-supercyclicity criterion on the same Hilbert space is equivalent to disjointness in supercyclicity of the corresponding left multiplication operators induced by T, ⋯, T in the strong operator topology. Besides, by the similar discussion, we also obtain that d-hypercyclicity criterion for any T, ⋯, T on the same Hilbert space is equivalent to d-hypercyclicity of the corresponding left multiplication operators induced by T, ⋯, T in the |.| topology. [ABSTRACT FROM AUTHOR]

We consider a splitting-based approximation of the abstract differential Riccati equation in the setting of Hilbert-Schmidt operators. The Riccati equation arises in many different areas and is important within the field of optimal control. In this paper we conduct a temporal error analysis and prove that the splitting method converges with the same order as the implicit Euler scheme, under the same low regularity requirements on the initial values. For a subsequent spatial discretization, the abstract setting also yields uniform temporal error bounds with respect to the spatial discretization parameter. The spatial discretizations commonly lead to large-scale problems, where the use of structural properties of the solution is essential. We therefore conclude by proving that the splitting method preserves low-rank structure in the matrix-valued case. Numerical results demonstrate the validity of the convergence analysis. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce and study multilinear pseudo-differential operators on Zn and Tn = (Rn/ 2πZn) the n-torus. More precisely, we give sufficient conditions and sometimes necessary conditions for Lp-boundedness of these classes of operators. L2-boundedness results for multilinear pseudo-differential operators on Zn and Tn with L2-symbols are stated. The proofs of these results are based on elementary estimates on the multilinear Rihaczek transforms for functions in L2(Zn) respectively L2(Tn) which are also introduced. We study the weak continuity of multilinear operators on the m-fold product of Lebesgue spaces Lpj(Zn), j = 1,...,m and the link with the continuity of multilinear pseudo-differential operators on Zn. Necessary and sufficient conditions for multilinear pseudo-differential operators on Zn or Tn to be a Hilbert-Schmidt operators are also given. We give a necessary condition for a multilinear pseudo-differential operators on Zn to be compact. A sufficient condition for compactness is also given. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we describe quantizations in the monoidal categories of unitary representations of compact connected Lie groups. For the -dimensional torus we show that the set of quantizations is isomorphic to the -dimensional torus. For connected compact Lie groups of rank , we get the result that the set of extendible quantizations of -modules is isomorphic to the set of quantizations of its maximal torus invariant under action by its Weyl group. For all these cases we give explicit formulae for quantizations and apply these to quantize Hilbert–Schmidt operators. [Copyright &y& Elsevier]

Abstract: Continuous analogs of the strong Szegő limit theorem may be formulated in terms of operators of the form for , where denotes the operator of multiplication by a suitably restricted mvf (matrix-valued function) acting on the space of vvfʼs (vector-valued functions) f that meet the constraint with and denotes the orthogonal projection onto the space of entire vvfʼs of exponential type⩽T that are subject to the same summability constraint. In this paper we study these operators for a more general class of Δ of the form , where , are mvfʼs in the Wiener plus algebra. This paper extends an earlier paper [6] by replacing the assumption that is an inner mvf for some by the less restrictive assumption that the Hankel operator with symbol is compact. We show that is a trace-class operator, that exists and is independent of Q and R when and . An example which shows that may depend on Q and R if these commutation conditions are not in force is furnished. [Copyright &y& Elsevier]

We consider one-dimensional Schrödinger-type operators in a bounded interval with non-self-adjoint Robin-type boundary conditions. It is well known that such operators are generically conjugate to normal operators via a similarity transformation. Motivated by recent interests in quasi-Hermitian Hamiltonians in quantum mechanics, we study properties of the transformations and similar operators in detail. In the case of parity and time reversal boundary conditions, we establish closed integral-type formulae for the similarity transformations, derive a non-local self-adjoint operator similar to the Schrödinger operator and also find the associated 'charge conjugation' operator, which plays the role of fundamental symmetry in a Krein-space reformulation of the problem. [ABSTRACT FROM AUTHOR]

Pseudo-differential operators with operator-valued symbols on the Heisenberg group $${\mathbb{H }}^n$$ are introduced. We give necessary and sufficient conditions on the symbols for which these operators are in the Hilbert-Schmidt class. These Hilbert-Schmidt operators are then identified with Weyl transforms with symbols in $$L^2({\mathbb{R }}^{2n+1}\times {\mathbb{R }}^{2n+1}).$$ We also give a characterization of trace class pseudo-differential operators on the Heisenberg group $${\mathbb{H }}^n$$ . A trace formula for these trace class operators is presented. [ABSTRACT FROM AUTHOR]