Chapter 1: A Riesz-Schauder Theory for Operator Polynomials.
This paper will be largely concerned with the properties of rational
functions with coefficients which are partial differential operators; the
Laplace transform calculus will follow from these properties in a standard
way once the properties themselves have been established. It is a useful
fact that the Riesz-Schauder theory on the spectra of compact operators may
be generalised to include the inverses of polynomials P(A,A) with compact
operator coefficients. The various elements of such a theory are all almost
immediate consequences of known results on perturbation and analytic families
of linear operators. I will only require these theorems for continuous
operators; however some results will be stated more generally for closed
operators. The main result is a theorem stating conditions under which the
poles are countable and isolated except at one point and that the index is 0
for every A f 0. This generalizes the main points of the Riesz-Schauder
theory to operator polynomials.
For their special case Visik and Agranovich [4] have already observed
the possibility of such a theory. In the context of Banach algebras Blum [7]
and Glickfeld [15], [16] have discussed rational functions of Banach algebra
elements. For algebras of commuting operators on Banach spaces Taylor [33],
[34] has developed a theory of joint spectra which allows factorization of
polynomials. Taylor has commented that this theory might be generalised to
the noncommutative case [34], [35]; it seems to me that such a generalisation
would help to illuminate the results of this paper. However it has not as yet
been carried out.
Definition 1.0. Let X be a Banach space and A = (A ,A ,... ,A ) be a
m
m-JAi
collection of closable operators Ai : X - X Vi. Let P(A,A) = \ \
be a polynomial of degree m in A e $ and a closed operator with domain
4
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