Given an operator A on a Hilbert space H and an analytic function f
defined in a neighborhood of o(A) , the spectrum of A , the operator f(A) is
defined via the Cauchy integral of the operator valued function f(z)(z - A)" 1
around a suitably chosen system of contours enclosing c(A) . This mapping f
- f(A) is an algebraic homomorphism from the algebra of all analytic
functions defined in some neighborhood of a(A) into the double commutant
of A such that 1 is mapped to the identity operator and z is mapped to A .
Moreover this map has a certain continuity property which makes it unique.
For details, the reader can consult pages 203-210 of [12], where this is
worked out in the framework of Banach algebras.
This mapping f f(A) is called the Riesz functional calculus or the
Riesz-Dunford functional calculus. The first appearance of these ideas is
[27], where only compact operators are considered and the only analytic
function considered is the characteristic function of an isolated point of the
spectrum. Though the topic, with the near simultaneous appearance of [16],
[24], and [30], all of which extended Riesz's ideas, takes on all the aspects of
one whose time had come, it is the work of Dunford [16] which is the most
complete in its treatment. In particular, it was Dunford who first proved the
Spectral Mapping Theorem. For this reason it is the custom of many,
including the authors, to call this the Riesz-Dunford functional calculus.
Whereas in the discussion of the Riesz-Dunford functional calculus the
idea is to fix the operator A and let f vary through a collection of analytic
functions, the attitude taken here is to fix the analytic function f and allow
the operator A to vary through the collection of all operators for which it
makes sense to define f(A) . Specifically, for an arbitrary (not necessarily
Previous Page Next Page