Summary: EXPLICIT FORMULAE FOR
TAYLOR'S FUNCTIONAL CALCULUS
Abstract. In this paper integral formulae, based on Taylor's functional calculus for
several operators, are found. Special cases of these formulae include those of Vasilescu
and Janas, and an integral formula for commuting operators with real spectra.
x1 Introduction. Let a = (a 1 ; : : : ; an ) be a commuting tuple of bounded linear
operators on a complex Banach space X,
let\Omega be an open subset containing the
Taylor's joint spectrum of a, Sp(a; X), and let f be a holomorphic function defined
on \Omega\Gamma Then Taylor [T] defined f(a) as follows:
f(a)x = 1
(R ff(z) f(z)x)dz;
for each x 2 X.
However, since R ff(z) is a homomorphism of cohomology, this means, in contrast
with the DunfordSchwartz calculus, that the functional calculus of Taylor is rather