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Characteristic function; invariant subspaces; biholomorphic automorphisms; functional model; coincidence

The characteristic function has been an important tool for studying completely non-unitary contractions on Hilbert spaces. In this note, we consider completely non-coisometric contractive tuples of commuting operators on a Hilbert space H. We show that the characteristic function, which is now an operator-valued analytic function on the open Euclidean unit ball in ℂⁿ, is a complete unitary invariant for such a tuple. We prove that the characteristic function satisfies a natural transformation law under biholomorphic mappings of the unit ball. We also characterize all operator-valued analytic functions which arise as characteristic functions of pure commuting contractive tuples.

T Bhattacharyya¹ J Eschmeier² J Sarkar¹

1. Department of Mathematics, Indian Institute of Science, Bangalore - 560 012, India
2. Fachbereich Mathematik, Universität des Saarlandes, Saarbrücken - 66123, Germany

Published

August 2006

Manuscript received

30 September 2005

Manuscript revised

7 April 2006