On the directional growth of the resolvent norm

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 $z,z^{\prime }\in \rho (A)$ let $[z,z^{\prime }]$ denote the straight line segment from $z$ to $z^{\prime }$. For each $z\in \rho (A)$ we classify the behavior of the resolvent norm $\zeta \mapsto \Vert R_A(\zeta )\Vert$ near $z$. Either there are $z^{\prime }\in \rho (A)$, $z^{\prime }\ne z$, $[z,z^{\prime }]\subset \rho (A)$, such that $\Vert R_A(\zeta )\Vert \ge \Vert R_A(z)\Vert +C|\zeta -z{|}^{\delta }$ for $\zeta \in [z,z^{\prime }]$ with $\delta =1$ or $\delta =2$, or the function $\zeta \mapsto \Vert R_A(\zeta )\Vert$ has a global minimum at $\zeta =z$.

7 pages