On the representation and approximation of a class of operator-valued analytic functions

Abstract

1. Main results. Let H be a separable Hilbert space, and a, b9 real and finite. Suppose that T(z) is, for each z O [a, b], a bounded operator-valued analytic function which satisfies the condition (R) For each (j> G H and z O [a, b], (r(z)0, 0) maps the upper halfplane into itself and the lower half-plane into itself We then call T(z) an R-operator, and the class of all /?-operators with the cut [a, b] will be called W[a, b]. The class (jj[a, b] generalizes to operators of the class R[a, b] of R-functions, those analytic functions on the cut complex plane which preserve upper and lower half-planes. For any f(z) G R[a, b], there is a nonnegative number a, a real number |8, and a positive measure JU with support on [a, b], such that