On the directional growth of the resolvent norm

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