Spectral properties of operator matrices on Banach spaces

Abstract

This doctoral dissertation has as its aim to present various results related to different types of invertibility of block operator matrices, whose some of the entries are known, and the others are unknown. Block matrices in question are upper triangular operator matrices, that is matrices whose entries below the main diagonal are zero operators. We will always assume that diagonal elements of such matrices are given, while elements above the main diagonal are not. It is understood that all operator matrices in question act on a direct topological sum of Banach spaces, as it is precised in the sequel. Our goal is to present characterization results for different types of invertibility of such operators, which then yield appropriate perturbation results and some "filling in holes" results. So far, investigations of this kind were mainly undertaken in the context of separable Hilbert spaces. Thus, countable orthogonal bases were frequently used in such research. The author of this dissertation has taken a different path. Instead of using linear bases of Banach spaces that need not be countable, the present author has rather worked using appropriate embedding mappings between certain subspaces of Banach spaces that have a topological complement. Moreover, specialist in this area have usually examined the case of 2×2 upper triangular operator matrices, while the author of this dissertation examines upper triangular operator matrices of an arbitrary dimension. In this way, the technique of Banach space embeddings introduced by Dragan S. Djordjevi´c in [12] is generalized to upper triangular operator matrices of an arbitrary dimension, and techniques from [54],[55] are adapted for operator matrices acting on a direct sum of spaces which need not be separable. The latter ideas represent the original scientific contribution of the present author, and they can be found in papers [44]{[48]