Unbound linear operators in operator ranges

Labuschagne, L. E.

Unbound linear operators in operator ranges

Master Thesis

1986

Abstract

Many results in operator theory for example some perturbation results, are at present known only in the Banach space case. The aim of this work is to provide a natural generalisation of such results by considering operator ranges (the image of a bounded operator defined everywhere on a Banach space) as well as investigating and characterizing some of the properties of operator ranges. For the sake of generality we will for the most part be considering unbounded or closed linear operators instead of continuous everywhere defined linear operators. We will not be attempting to give exhaustive coverage of unbounded linear operators but will try to give some insight into the use of operator range techniques in the theory of unbounded linear operators. The first chapter will be aimed mainly at defining and introducing concepts used in later chapters. In the second chapter we turn our attention to the conjugate of a linear operator whilst also briefly looking at projections in an operator range. Chapter three is concerned mainly with investigating and characterizing the closed range property of linear operators whereas in the first part of chapter four we will be proving some fairly well known results on compact, precompact and strictly singular operators to be used in chapter five. In the second half of chapter four we will investigate the relationship between weakly compact operators and pre-reflexive spaces. Chapter five will be dealing with perturbation of semi-Fredholm operators by first of all continuous and then by strictly singular operators. We close with a discussion of the instability of non-semi-Fredholm operators under compact and a-compact perturbations.

Keywords

mathematics and applied mathematics

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