The effect of perturbations of linear operators on their polar decomposition

TL;DR

This paper investigates how perturbations affect the polar decomposition of operators, extending known matrix results to infinite-dimensional operators and establishing stability conditions for the partial isometry component.

Contribution

It proves the stability of the partial isometry in the polar decomposition under perturbations for infinite-dimensional operators, with weaker conditions than previously known for matrices.

Findings

Partial isometry stability under perturbations in infinite-dimensional spaces

Weaker conditions than equal-rank for matrix perturbations

Range and kernel proximity imply stability in the gap metric

Abstract

The effect of matrix perturbations on the polar decomposition has been studied by several authors and various results are known. However, for operators between infinite-dimensional spaces the problem has not been considered so far. Here, we prove in particular that the partial isometry in the polar decomposition of an operator is stable under perturbations, given that kernel and range of original and perturbed operator satisfy a certain condition. In the matrix case, this condition is weaker than the usually imposed equal-rank condition. It includes the case of semi-Fredholm operators with agreeing nullities and deficiencies, respectively. In addition, we prove a similar perturbation result where the ranges or the kernels of the two operators are assumed to be sufficiently close to each other in the gap metric.