Remarks on the convergence of pseudospectra

TL;DR

This paper proves the convergence of pseudospectra in Hausdorff distance for certain operators in Hilbert spaces and characterizes when the resolvent norm is constant, extending previous results and providing illustrative examples.

Contribution

It establishes pseudospectra convergence under general conditions and characterizes constant resolvent norm behavior, broadening the class of operators understood.

Findings

Pseudospectra converge in Hausdorff distance for operators in different Hilbert spaces.

Operators with constant resolvent norm on an open set have that constant as the global minimum.

Examples demonstrate various resolvent norm behaviors and applicability of the new characterization.

Abstract

We establish the convergence of pseudospectra in Hausdorff distance for closed operators acting in different Hilbert spaces and converging in the generalised norm resolvent sense. As an assumption, we exclude the case that the limiting operator has constant resolvent norm on an open set. We extend the class of operators for which it is known that the latter cannot happen by showing that if the resolvent norm is constant on an open set, then this constant is the global minimum. We present a number of examples exhibiting various resolvent norm behaviours and illustrating the applicability of this characterisation compared to known results.