On the directional growth of the resolvent norm

TL;DR

This paper investigates the local behavior of the resolvent norm of a closed operator on a Hilbert space, classifying whether it grows linearly, quadratically, or has a minimum at a point.

Contribution

It provides a classification of the growth behavior of the resolvent norm near points in the resolvent set of an operator.

Findings

The resolvent norm either grows at least linearly or quadratically along certain segments.

Alternatively, the resolvent norm can have a global minimum at a point.

The classification depends on the local behavior near points in the resolvent set.

Abstract

Let $A$ be a closed densely defined operator on a separable Hilbert space $\mathcal{H}$. Assume the resolvent set $\rho(A)$ is non-empty. For $z,z'\in\rho(A)$ let $[z,z']$ denote the straight line segment from $z$ to $z'$. For each $z\in\rho(A)$ we classify the behavior of the resolvent norm $\zeta\mapsto\lVert R_A(\zeta) \rVert$ near $z$. Either there are $z'\in\rho(A)$, $z'\neq z$, $[z,z']\subset\rho(A)$, such that $\lVert R_A(\zeta) \rVert \geq \lVert R_A(z) \rVert + C\lvert \zeta-z \rvert^\delta$ for $\zeta\in[z,z']$ with $\delta=1$ or $\delta=2$, or the function $\zeta\mapsto\lVert R_A(\zeta) \rVert$ has a global minimum at $\zeta=z$.