A local directional growth estimate of the resolvent norm

TL;DR

This paper investigates the growth behavior of the resolvent norm of certain linear operators on Hilbert spaces, establishing quadratic growth in specific directions and providing criteria for local minima, with implications for non-normal operators.

Contribution

It introduces new directional growth estimates for the resolvent norm and offers criteria for local minima, extending understanding of resolvent behavior without relying on the maximum principle.

Findings

Resolvent norm grows at least quadratically in certain directions.

The resolvent norm cannot have local maxima for the studied class.

Criteria for the existence of local non-degenerate minima are established.

Abstract

We study the resolvent norm of a certain class of closed linear operators on a Hilbert space, including unbounded operators with compact resolvent. It is shown that for any point in the resolvent set there exist directions in which the norm grows at least quadratically with the distance from this point. This provides a new proof not using the maximum principle that the resolvent norm of the considered class cannot have local maxima. Finally, we give new criteria for the existence of local non-degenerate minima of the resolvent norm and provide examples of (un)bounded non-normal operators having this property.