A Short Proof of an Interpolation Theorem

Abstract

In this note we give a simple proof of an operator-interpolation theorem (Theorem 2) due originally to Donoghue [6], and Lions-Foias [7].Let be the complex plane, the open upper half-plane, the real line, ℛ+ and ℛ- the non-negative and non-positive axes. Denote by the class of positive functions on which extend analytically to —ℛ-, and map into itself. Denote by ’ the class of functions φ such that φ(x1/2)2 is in .