Best approximation by diagonal compact operators

TL;DR

This paper investigates the existence and properties of compact Hermitian operators that best approximate a given operator in the operator norm, focusing on diagonal compact operators and their approximation characteristics.

Contribution

It characterizes conditions for the existence of such best approximations and provides examples of operators that do not attain the minimum approximation.

Findings

Characterization of compact Hermitian operators with minimal distance to diagonal compact operators

Existence criteria for best approximation by diagonal compact operators

Example of a trace class operator that does not attain the minimum approximation

Abstract

We study the existence and characterization properties of compact Hermitian operators C on a separable Hilbert space H such that ||C|| is less or equal than || C + D ||, for all D in D(K(H)). This property is equivalent to || C || = min{||C+D||: D in D(K(H))} = dist (C,D(K(H))), where D(K(H)) denotes the space of compact diagonal operators in a fixed base of H and ||.|| is the operator norm. We also exhibit a positive trace class operator that fails to attain the minimum in a compact diagonal.