Level sets of the resolvent norm of a linear operator revisited

TL;DR

This paper investigates the conditions under which the resolvent norm of linear operators on Banach spaces can be constant on open sets, revealing geometric constraints and constructing specific counterexamples.

Contribution

It establishes that constant resolvent norm on open sets is impossible in complex strictly convex spaces, and constructs examples in certain non-convex spaces, highlighting geometric influences.

Findings

Resolvent norm cannot be constant on open sets in complex strictly convex spaces.

Constructed examples of operators with constant resolvent norm in specific non-convex spaces.

Proved such examples cannot exist in co-dimension 1 spaces.

Abstract

It is proved that the resolvent norm of an operator with a compact resolvent on a Banach space $X$ cannot be constant on an open set if the underlying space or its dual is complex strictly convex. It is also shown that this is not the case for an arbitrary Banach space: there exists a separable, reflexive space $X$ and an unbounded, densely defined operator acting in $X$ with a compact resolvent whose norm is constant in a neighbourhood of zero; moreover $X$ is isometric to a Hilbert space on a subspace of co-dimension $2$. There is also a bounded linear operator acting on the same space whose resolvent norm is constant in a neighbourhood of zero. It is shown that similar examples cannot exist in the co-dimension $1$ case.