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 con- cerned with spectral analysis of the pair (A;B): We establish some properties concerning the spectrum of the linear operator pencils A 􀀀 B when B is not necessarily invertible and 2 C: Also, we use the functional calculus for the pair (A;B) to prove the corresponding spectral mapping theorem for (A;B): In addition, we de ne 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 opera- tors.