Capacity of elements of Banach algebras
Stirling, David G.
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As its name suggests, this thesis is an account of the recent theory of the capacity of elements of Banach algebras. The first chapter contains a summary of the background theory, other than fundamentals, used, and consists mainly of perturbation theory of linear operators and certain properties (Jf strictly singular operators. This chapter relies heavily on the work of T„ Kato, both in his own papers and the book by S. Goldberg "Unbounded Linear Operators". Chapter 2 introduces the notion of capacity, following Halmos in his paper "Capacity in Banach algebras", and several small new results are proved, and counterexamples given, to tidy up "loose ends". The question of the capacity of the sum of two quasialgebraic elements (i.e. ones with capacity zero) is raised, and a partial solution given. The perturbation theory of Chapter 1 is applied to show the equality of the capacity of the spectrum and the Fredholn spectrum of an operator on a Banach space, whence it is shown that if J is a closed two-sided ideal of B(x) containing only Riesz operators, then perturbation by an element of J leaves the capacity invariant; this is true, in particular, for compact operators. A converse theorem is proved for Hilbert space, Chapter 3 introduces the new concept of the joint capacity of an r-tuple of elements cf a commutative Banach algebra, and develops the theory of this notion, Much of the theory parallels, xn a weaker form, that of the original concept, but there are significant differences. Finally, a perturbation theorem, similar to the original one is proved for the joint capacity.