Hilbert-Schmidt Operators and Frames - Classification, Approximation by Multipliers and Algorithms

TL;DR

This paper explores the relationship between frames and Hilbert-Schmidt operators, providing criteria, constructions, and algorithms for approximation and analysis within this framework.

Contribution

It introduces new criteria for Hilbert-Schmidt operators using frames, constructs specific sequences for these operators, and develops an algorithm for optimal approximation by frame multipliers.

Findings

Criteria for Hilbert-Schmidt operators using frames

Construction of Bessel sequences, frames, and Riesz bases for these operators

Efficient computation of Hilbert-Schmidt inner products and approximation algorithms

Abstract

In this paper we deal with the connection of frames with the class of Hilbert Schmidt operators. First we give an easy criteria for operators being in this class using frames. It is the equivalent to the criteria using orthonormal bases. Then we construct Bessel sequences frames and Riesz bases for the class of Hilbert Schmidt operators using the tensor product of such sequences in the original Hilbert space. We investigate how the Hilbert Schmidt inner product of an arbitrary operator and such a rank one operator can be calculated in an efficient way. Finally we give an algorithm to find the best approximation, in the Hilbert Schmidt sense, of arbitrary matrices by operators called frame multipliers, which multiply the analysis coefficents by a fixed symbol followed by a synthesis.