By introducing the concept of -convertibility, we extend Nielsen’s and Vidal’s theorems to the entanglement transformation of infinite-dimensional systems. Using an infinitedimensional version of Vidal’s theorem we derive a new stochastic-LOCC (SLOCC) monotone which can be considered as an extension of the Schmidt rank. We show that states with polynomially-damped Schmidt coefficients belong to...

For any bipartite quantum system the Schmidt decomposition allows us to express the state vector in terms of a single sum instead of double sums. We show the existence of the Schmidt decomposition for tripartite system under certain condition. If the partial inner product of a basis (belonging to a Hilbert space of smaller dimension) with the state of the composite system gives a disentangled b...

This paper introduces a novel mathematical and computational framework, namely Log-Hilbert-Schmidt metric between positive definite operators on a Hilbert space. This is a generalization of the Log-Euclidean metric on the Riemannian manifold of positive definite matrices to the infinite-dimensional setting. The general framework is applied in particular to compute distances between covariance o...

We show that Hilbert–Schmidt operators can be used to define frame-like structures for $L^2(\mathbb{R}^d )$ over lattices in $\mathbb{R}^{2d} $ include multi-window Gabor frames as a special case. These are called g-frames

In this paper we consider products of composition and differentiation operators on the Hardy spaces. We provide a complete characterization boundedness compactness these operators. M. Moradi Fatehi [1] obtain explicit condition for to be Hilbert-Schmidt have theoretical application with series perfect symbols.

Abstract We characterize the Hilbert–Schmidt class membership of commutator with Hilbert transform in two weight setting. The characterization depends upon symbol being a new weighted Besov space. This follows from Schatten S p result for dyadic paraproducts, where $1&lt; &lt; \infty $ 1 &lt; p</...

Nonlinear vector integral equations are considered. Solution estimates and solvability conditions are derived. Applications to the periodic boundary value problem are also discussed. Under some restrictions our results improve the well-known ones. The main tool in the paper is the recent estimates for the resolvent of Hilbert-Schmidt operators.

The purpose of this paper is to give a survey of the progress, advantages and limitations of various operator inequalities involving improved Young's and its reverse inequalities related to the Kittaneh-Manasrah inequality. We also present our new progress to the related research topics. New scalar versions of Young's inequalities are promoted, the operator version and the Hilbert-Schmidt form ...

We study the stability properties of nonlinear multi-task regression in reproducing Hilbert spaces with operator-valued kernels. Such kernels, a.k.a. multi-task kernels, are appropriate for learning problems with nonscalar outputs like multi-task learning and structured output prediction. We show that multi-task kernel regression algorithms are uniformly stable in the general case of infinite-d...

To help understand various reproducing kernels used in applied sciences, we investigate the inclusion relation of two reproducing kernel Hilbert spaces. Characterizations in terms of feature maps of the corresponding reproducing kernels are established. A full table of inclusion relations among widely-used translation invariant kernels is given. Concrete examples for Hilbert-Schmidt kernels are...