Perturbations of discrete spectra of holomorphic operator-valued functions

Abstract

Let X be a complex Banach space and L(X) be the Banach space of all bounded linear operators on X. Let $$\Omega \subset {{\mathbb {C}}}$$ be open and connected. Let $$T, V : \Omega \longrightarrow L(X)$$ be holomorphic operator-valued functions. We consider the one parameter family of operator-valued functions $$W(\lambda , t) := T(\lambda ) + t V(\lambda )$$, for $$t \in {{\mathbb {C}}}$$, and analyze evolution of the discrete eigenvalues of $$W(\lambda , t)$$ when t varies in $${{\mathbb {C}}}.$$ We provide a brief review of the discrete spectrum of $$T(\lambda )$$ and present several equivalent characterizations for discrete eigenvalues of $$T(\lambda ).$$ We also prove Rouche’s theorem for operator-valued functions under a weaker assumption, which we utilize to derive perturbation bounds for the discrete eigenvalues of $$W(\lambda , t)$$ when |t| is small.