On the Taylor functional calculus
by
V. Müller
Institute of Mathematics of the Czeck Academy

Let A=(A₁, ..., A_n) be an n-tuple of mutually commuting operators acting on a Banach space X. The existence of the Taylor functional calculus [T1], [T2] is one of the most important results of spectral theory. However, the formula defining f(A) for a function f analytic on a neighbourhood of the Taylor spectrum has some drawbacks. The operator f(A) is defined locally, the formula gives only f(A)x for each x in X. Therefore it is not easy to see that f(A) is bounded. Moreover, the formula is rather inexplicit and it is quite difficult to prove even the basic properties of the calculus.

The situation is better for Hilbert space operators. In [V1], [V2] Vasilescu gave an explicit Martinelli-type formula defining f(A) which is much easier to handle.

The ideas of Vasilescu were used in [KM] to prove a similar formula for Banach space operators. The method works, however, only for functions analytic on a neighbourhood of the split-spectrum which is in general bigger than the Taylor spectrum. The main tool is the existence of generalized inverses for operators that appear in the Koszul complex.

The aim of the talk will be to obtain a similar formula for the general Taylor functional calculus. The main innovation is the use of non-linear (but continuous) general inverses. In this way we obtain a formula that defines f(A) globally, and so the continuity of f(A) and the continuity of the functional calculus become clear. The formula is more explicit and so it is possible to avoid some technical difficulties in the proof of the basic properties of the calculus. The cohomogical methods are avoided and the proofs are based only on the Stokes and the Bartle-Graves theorems.

Date received: June 15, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caeo-72.