FUNCTIONAL CALCULI FOR HILBERT SPACE OPERATORS

Abstract

Let T be a bounded operator on a Hilbert space H with Von Neumann spectral set X. If there exists no non-zero reducing subspace of H restricted to which T is a normal operator with spectrum contained in the boundary of X and if the uniform algebra R(X) is pointwise boundedly dense in H^∞ (X°), then there exists a functional calculus f → f(T) for f ε H^∞ (X°). A similar result for the two-variable case is also proved.