Essential spectra of elementary operators

Abstract

Abstract. This paper describes the essential spectrum and index function of the operator X -» AXB, where A, B, and X are Hubert space operators. Analogous results are given for the restriction of this operator to a norm ideal and partial analogues are given for sums of such operators and for the case when the operators act on a Banach space.1. Introduction. The purpose of this note is to describe the Fredholm essential spectrum and index function for a class of operators of the form X -» AXB, where A, B, and X are Hilbert space operators. We also describe the essential spectra and index functions of the restrictions of these operators to norm ideals. These results thus complement the spectral analysis of multiplications initiated by Lumer and Rosenblum [20].Let % denote a separable, infinite-dimensional, complex Hilbert space and let £,(%) denote the algebra of all bounded linear operators on %. For operators A and B in £(%), let S = S (A, B) and <3" = <5(A, B) denote the operators on &(%)

Weak essential spectra of multiplication operators on spaces of bounded linear operators

“…Our study complements several previous results concerning the spectrum of A A B due to Lumer and Rosenblum [14], Brown and Pearcy [2] and Schechter [19] as well as the essential spectrum due to Fialkow [6] and Eschmeier [5]. We refer to the surveys of Curto [3] and Fialkow [7] for further results and references.…”

“…the spectral formulas for such restrictions on I(dP), see [2,19,6]. the spectral formulas for such restrictions on I(dP), see [2,19,6].…”

“…This contradicts the assumption that 41 C trw(At). Identity (6) implies that RF(x ~ | Yk) = x ~ | Yk + 1;1(x ~ | and hence (12) (y'k, We commence by establishing (9) with the help of suitable evaluations.…”

Tensor products and elementary operators.

“…This result together with a spectral mapping theorem allows us to prove, that [13] and [15] in the positive sense. This result together with a spectral mapping theorem allows us to prove, that [13] and [15] in the positive sense.…”

A remark on the range of elementary operators

“…Given A, B ∈ L(H), we define the elementary operator ∆ A,B as If A = B, we write simply ∆ A for ∆ A,A . The properties of elementary operators, their spectrum (see [9], [10], [12]), norm ( [15], [17] and [18]) and ranges ( [1], [2], [3], [4], [6], [12], [13], [14], and [16]) have been studied intensively, but many problems remain open [12].…”