Completely Positive Maps Induced by Composition Operators
نویسنده
چکیده
We consider the completely positive map on the Toeplitz operator system given by conjugation by a composition operator; that is, we analyze operators of the form C∗ φTfCφ We prove that every such operator is weakly asymptotically Toeplitz, and compute its asymptotic symbol in terms of the Aleksandrov-Clark measures for φ. When φ is an inner function, this operator is Toeplitz, and we show under certain hypotheses that the iterates of Tf under a suitably normalized form of this map converge to a scalar multiple of the identity. When φ is a finite Blaschke product, this scalar is obtained by integrating f against a conformal measure for φ, supported on the Julia set of φ. In particular the composition operator Cφ can detect the Julia set by means of the completely positive map.
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