Clark measures and operators similar to a contraction

Abstract

In this article the authors study the conditions under which a certain operator is similar to a contraction. The specific question was originally posed by R. Douglas in the 1990s, and is an offshoot to a question posed by P. Halmos in the 1970s. Halmos' question arose due to Von Neumann's inequality which says that for any contraction T and any polynomial p, It follows directly from this inequality that contractions are polynomially bounded. In addition, it can be shown that if A is a bounded operator similar to a contraction, and B(z) is a Blaschke product such that the closure of the set of poles of B lies off the spectrum of A, then B(A) is also similar to a contraction. The question R. Douglas posed is whether the converse is true, that is, given the above conditions, if B(A) is similar to a contraction, is A similar to a contraction? T. Lance and M. Stessin proved it to be the case where B is a finite Blaschke product, using a method that exploits the Wold Decomposition of the Hardy Space . Later, Stessin expanded the result to include a certain class of infinite Blaschke products. Integral to the method of proof is the estimate of a particular operator related to the Wold Decomposition. The upper and lower estimates suggested a connection between (Clark measure), and the Lebesgue measure of A, where A is an arc of the unit circle about an accumulation point of Eω (the carrier set for the Clark measure). We explore this connection and see how it plays a role in widening the class of inner functions for which Douglas' question is answered affirmatively.