Vol. 60, No. 2, 1975

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

Stephen LaVern Campbell

Vol. 60 (1975), No. 2, 37–49

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 T1 and spectral radius strictly less than one. Let V T be its Potapov inner function and define UT = V TV T(1). The structure of nonnormal T for which UT 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 TT and T + T commute is examined.

Mathematical Subject Classification 2000
Primary: 47A45
Received: 16 May 1974
Published: 1 October 1975
Stephen LaVern Campbell