Contemporary Mathematics Volume 186, 1995 EXPLICIT FORMULAE FOR TAYLOR'S FUNCTIONAL CALCULUS D. W. ALBRECHT 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. §1 Introduction. Let a = (a1,... , an) be a commuting tuple of bounded linear operators on a complex Banach space X, let n be an open subset containing the Taylor's joint spectrum of a, Sp(a, X), and let f be a holomorphic function defined on 0. Then Taylor [T] defined f(a) as follows: f(a)x = (2:i)n i (Ro:(z)f(z)x)dz, for each x EX. However, since Ro:(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 spectrum, i.e. Sp(a, X) C IRn (3) n is a bounded convex domain given by a smooth defining function. §2 Notation. Let (J = (s1 , ... , sn) and dz = (dz1, ... , dzn) denote tuples of inde- terminates, and A[(J U dz] denote the exterior algebra, over C, generated by these tuples. Let AP[(J U dz] denote the subspace of p-forms in A[(J U dz]. For 1 :S i :S n, let si be operators on A[(J u dz] defined by Si(~) = Si 1\ ~. for every~ E A[(J u dz]. We shall call these the creation operators for (J on A[(J U dz]. Let dz1,... , dzn also denote the creation operators for dz on A[(J U dz], and for each 1 :S i :S n, let Si denote the operators on A[(J u dz] which satisfy the following anti-commutation relations: s s + s s = o sis + s si = { ~ if i = j ifi:f j' 1991 Mathematics Subject Classification. Primary 47A60 Secondary 47A56. The detailed version of this paper has been submitted for publication elsewhere 1 © 1995 American Mathematical Society 0271-4132/95 $1.00 + $.25 per page
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