Quasi-nilpotent perturbations of the generalized Kato spectrum

In this paper, we show that the generalized Kato spectrum of a bounded operator in a Banach space is invariant under perturbation by commuting quasi-nilpotent operators, and the Kato spectrum is stable under additive commuting nilpotent perturbations. Our results are used to give an equivalent definition of the generalized Kato spectrum.

Generalized Kato linear relations

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.