## Capacity of elements of Banach algebras

##### Abstract

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.