Locating the spectrum of a relation matrix through those of its inputs

Abstract

Let [Formula: see text] and [Formula: see text] be Banach spaces. When [Formula: see text] and [Formula: see text] are linear relations in [Formula: see text] and [Formula: see text], respectively, we denote by [Formula: see text] the linear relation matrix acting on [Formula: see text] of the form [Formula: see text], where [Formula: see text] is the zero operator and [Formula: see text] is a bounded operator from [Formula: see text] to [Formula: see text]. In this paper, we prove that if [Formula: see text] denotes the Weyl spectrum, the Browder spectrum or the Drazin spectrum of a linear relation, then for every [Formula: see text] we have the equality [Formula: see text] where [Formula: see text] a subset of [Formula: see text]. Moreover, we explore how Weyl’s theorem and Browder’s theorem hold for linear relation matrices.