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Title: Non-Compact Operators in Banach Spaces.
Author: Potter, A. J. B.
ISNI:       0000 0001 3496 8704
Awarding Body: University of Sussex
Current Institution: University of Sussex
Date of Award: 1972
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In this thesis we show that many well known results in the theory of compact operators in Banach spaces can be generalised in such a way that the compactness condition is replaced by a less restrictive assumption. The two particular branches of the theory of compact operators, with which we are concerned, are the theories of positive compact operators acting in partially ordered Banach spaces and of analytic type compact operators. In Chapter 2 we extend many of the important theorems concerned with positive compact operators to k-set contractions and other classes of non-compact operators such as the k-ball contractions and P-compact mappings. We deal with linear, homogeneous and non-linear mappings. Chapter 3 contains a development of the theory of analytic type A-proper mappings which parallels Browder's theory of analytic type compact operators (see [ 7 ]). From this we deduce theorems on the structure of solution sets to various operator equations and, in particular, we obtain uniqueness theorems for k-ball contractions (k < 1). The first chapter is merely a survey of the existence theory we require in the rest of the thesis. None of the results therein are original although we give independent proofs of two important theorems which we feel may have some mathematical interest. In the final chapter we give Borne applications of our results to the theories of differential and integral equations.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available