On the Essential Spectra of Quasisimilar Operators

Abstract

B. Sz.-Nagy and C. Foiaş gave the first examples of quasisimilar operators with different spectra [25]. Indeed, quasisimilar operators can even have different spectral radius [19] (see also [15]). Nevertheless, T. B. Hoover has shown in [19] that if T and S are quasisimilar, then σ(T)∩σ(S) ≠ ∅, where σ(R) denotes the spectrum of the (bounded linear) operator R. In [12], the author improved Hoover's result by showing that each component of σ(S) intersects σ(T), and viceversa. This, in turn, was further improved by L. A. Fialkow in [7]. (Actually, Fialkow's results were obtained independently of [12].)L. A. Fialkow [7] and L. R. Williams [27] independently proved that σe(T)∩σe(S) ≠ ∅, where σe(R) denotes the essential spectrum of R. Several authors have raised the following natural question:Is it also true that each component of σe(S) intersects σe(T), and viceversa? [7] [21] [26].