On the spectrum of the Sylvester-Rosenblum operator acting on triangular algebras

Abstract

Let A and B be algebras and M be an A -B-bimodule. For A ∈ A , B ∈B, we define the Sylvester-Rosenblum operator τA,B :M →M via τA,B(M) = AM+MB for all M ∈ M . We investigate the spectrum of τA,B in three settings, namely: (a) when A = B = Tn(F) , the set of upper-triangular matrices over an algebraically closed field F and M ⊆ Mn(F); (b) when A = B =M is a unital triangular Banach algebra; and (c), when M = T (N ) is the nest algebra associated to a nest N on a complex, separable Hilbert space and A = B = CI+K (N ) consists of the unitization of the algebra of compact operators in T (N ) .