Operator-valued inner functions analytic on the closed disc. II

Stephen LaVern Campbell

Vol. 60 (1975), No. 2, 37–49

DOI: 10.2140/pjm.1975.60.37

Abstract

An operator-valued inner function V is called scalar if {V (w) : |w| < 1} is a commuting family of normal operators. Suppose that T is a bounded linear operator with ∥T∥≦ 1 and spectral radius strictly less than one. Let V _T be its Potapov inner function and define U_T = V _TV _T^∗(1). The structure of nonnormal T for which U_T is scalar is discussed. An explicit characterization is given if the underlying Hilbert space is finite dimensional. Examples are given for the infinite dimensional case. The relationship between scalar inner functions and operators for which T^∗T and T^∗ + T commute is examined.

Mathematical Subject Classification 2000

Primary: 47A45

Milestones

Received: 16 May 1974

Published: 1 October 1975