Differentiation and Covering Constants for Hilbert-Schmidt and Quasi-Hilbert-Schmidt Operators

TL;DR

This paper investigates the derivatives and covering constants of Hilbert-Schmidt and quasi-Hilbert-Schmidt operators, establishing key properties and differentiability results for these classes of operators on separable Hilbert spaces.

Contribution

It introduces the concept of quasi-Hilbert Schmidt operators and computes their derivatives, extending the understanding of operator differentiability and covering constants.

Findings

Covering constant for Hilbert-Schmidt operators is zero.

Differentiability of Hilbert-Schmidt integral operators is analyzed.

Example of quasi-Hilbert Schmidt operators with derivatives provided.

Abstract

In this paper, we calculate the Frechet derivatives and Mordukhovich derivatives (or coderivatives) of Hilbert Schmidt operators on separable Hilbert spaces, by which we prove that the covering constant for Hilbert-Schmidt operators is zero. As an important class of Hilbert Schmidt operators, we study the differentiability of Hilbert Schmidt integral operators. Then, we introduce the concept of quasi-Hilbert Schmidt operators on separable Hilbert spaces. We provide an example of quasi-Hilbert Schmidt operators and find its Frechet derivatives and Mordukhovich derivatives.