Numerical and essential numerical ranges on $\ell_p$

Abstract

The paper offers the first systematic study of ordinary and essential numerical ranges of operators on $\ell_p$, $1<p<\infty$, as an atomic picture within a broader $L^p$ project. The paper begins with Banach-space foundations, including the finite-codimensional description of the essential numerical range and a Banach-space convex-hull inclusion for the essential spectrum. It then turns to finite-dimensional $\ell_p$ geometry, where one finds both positive star-shapedness phenomena and explicit $2\times2$ counterexamples. On $\ell_p$, we prove that the essential numerical range is compact and convex, identify it with the algebraic numerical range of the Calkin image, obtain a compact-perturbation formula, and show that, moreover, the closure of the numerical range is star-shaped, while points in the interior of the essential numerical range are exact star-centres of the numerical range. The paper illustrates the developed theory with sequence-space examples, covering tridiagonal Toeplitz operators and the discrete Hilbert transform, and, after relating our study to a variant of the Crouzeix inequality, closes with a brief discussion of extensions to spaces of class $(P)$ and to joint essential numerical ranges.