Local spectral theory for 2×2 operator matrices

Abstract

We discuss the spectral properties of the operator MC∈ℒ(X⊕Y) defined by MC:=(AC0B), where A∈ℒ(X), B∈ℒ(Y), C∈ℒ(Y,X), and X, Y are complex Banach spaces. We prove that (SA∗∩SB)∪σ(MC)=σ(A)∪σ(B) for all C∈ℒ(Y,X). This allows us to give a partial positive answer to Question 3 of Du and Jin (1994) and generalizations of some results of Houimdi and Zguitti (2000). Some applications to the similarity problem are also given.