Generalized Kato linear relations

Abstract

For a Banach space the notions of generalized Kato linear relation and the corresponding spectrum are introduced and studied. We show that the symmetric difference between the generalized Kato spectrum and the Goldberg spectrum of multivalued linear operators in Banach spaces is at most countable. The obtained results are used to describe the generalized Kato spectrum of the inverse of the left shift operator regarded as a linear relation.