Circles in the spectrum and the geometry of orbits: a numerical ranges approach

TL;DR

This paper explores the relationship between the spectrum of bounded linear operators on Hilbert spaces and their numerical ranges, establishing new spectral characterizations and convergence results through a numerical ranges approach.

Contribution

It introduces a novel connection between the unit circle in the spectrum and operator orbits, and generalizes Arveson's theorem using numerical ranges.

Findings

Characterization of the essential approximate point spectrum via operator orbits

Generalizations of Arveson's theorem using numerical ranges

Weak convergence of operator powers implies uniform convergence of compressions

Abstract

We prove that a bounded linear Hilbert space operator has the unit circle in its essential approximate point spectrum if and only if it admits an orbit satisfying certain orthogonality and almost-orthogonality relations. This result is obtained via the study of numerical ranges of operator tuples where several new results are also obtained. As consequences of our numerical ranges approach, we derive in particular wide generalizations of Arveson's theorem as well as show that the weak convergence of operator powers implies the uniform convergence of their compressions on an infinite-dimensional subspace. Several related results have been proved as well.