A Spectral Analysis of Linear Operator Pencils on Banach Spaces with Application to Quotient of Bounded Operators

Abstract

Let X and Y two complex Banach spaces and (A,B) a pair of bounded linear operators acting on X with value on Y. This paper is concerned with spectral analysis ofthe pair (A;B): We establish some properties concerning the spectrum of the linear operator pencils (A-lambda B) when B is not necessarily invertible and lambda is a complex number. Also, we use the functional calculus for the pair (A,B) to prove the corresponding spectral mapping theorem for (A,B). In addition, we define the generalized Kato essential spectrum and the closed range spectra of the pair (A,B) and we give some relationships between this spectrums. As application, we describe a spectral analysis of quotient operators.