Conjugating by singular operators: On the boundedness of similarity transforms near singular points

TL;DR

This paper investigates the boundedness of similarity transforms near singular points in operator theory, providing algebraic criteria for when such transformations remain bounded, especially in finite-dimensional Hilbert spaces.

Contribution

It offers a complete algebraic classification of operators for which similarity transforms remain bounded near singular points, extending to generalizations involving bounded linear maps.

Findings

Classified operators with bounded similarity transforms near singular points.

Identified algebraic criteria for coefficients in inverse operators.

Extended results to finite-dimensional Hilbert space contexts.

Abstract

We consider the question of, given operators $A$, $Z$ and a sequence of invertible operators $U_n\to Z$, whether the sequence $U_nAU_n^{-1}$ is bounded in norm, as well as generalizations of this where $U_nAU_n^{-1}$ is modified by some bounded linear map on bounded linear operators. In the setting of Hilbert spaces, we provide a complete classification in terms of algebraic criteria of those $A$ for which such a sequence exists, as long as $Z$ is of generalized index zero, which always holds in finite-dimensional contexts. In the process, we prove that particular coefficients arising in inverses of certain good paths going to $Z$ can also be classified in terms of an entirely algebraic criterion.