Essential norm resolvent estimates and essential numerical range

Abstract

The main result of this paper are novel two-sided estimates of the essential resolvent norm for closed linear operators $T$. We prove that the growth of $\Vert (T-\lambda {)}^{-1}{\Vert }_{\text{e}}$ is governed by the distance of a point $\lambda \in \rho (T)\setminus W_{\text{e}}(T)$ to the essential numerical range $W_{\text{e}}(T)$. We extend these bounds even to points $\lambda \in \mathbb{C}\setminus W_{\text{e}}(T)$ outside the resolvent set $\rho (T)$ with $(T-\lambda {)}^{-1}$ replaced by the Moore–Penrose resolvent $(T-\lambda {)}^{†}$. We use similar ideas to prove essential growth bounds in terms of the real part of the essential numerical range of generators of $C_0$-semigroups. Further, we study the essential approximate point spectrum ${\sigma }_{\text{eap}}(T)$ and the essential minimum modulus ${\gamma }_{\text{e}}(T)$, in particular, their relations to the various essential spectra and the essential norm of the Moore–Penrose inverse, respectively. An important consequence of our results are new perturbation results for the spectra and essential spectra (of type 2) for accretive and sectorial $T$. Applications e.g. to Schrödinger operators with purely imaginary rapidly oscillating potentials in ${\mathbb{R}}^d$ illustrate our results.