Explicit Formulae for Taylor's Functional Calculus

. 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 (2ßi) n Z \Omega (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 Dunford-Schwartz calculus, that the functional calculus of Taylor is rather inexplicit. In this paper we will show how explicit formulae for Taylor's functional calculus can be obtained, under certain circumstances which include the following: (1) X is a Hilbert space; (2) a has real spect..