On the functional calculus of contractions with nonvanishing unitary asymptotes (Q1174215)

Let \(T\) be a contraction on a Hilbert space \(H\). There exist a Hilbert space \(H'\), a unitary operator \(T'\) on \(H'\), and an operator \(X: H\to H'\) and such that \(T'X=XT\) and such that the triplet \((H',T',X)\) is `universal' with these properties. The author calls \(T'\) the unitary asymptote of \(T\). If \(T\) is an absolute continuous contraction then \(T'\) has absolutely continuous spectral measure, and hence there is a Borel set \(\Gamma\) such that arclength on \(\Gamma\) is a saclar measure for \(T'\). The main purpose of this article is to produce a new factorization technique in the theory of dual algebras. The author illustrates this technique by providing, along with some new results, a new proof of the following result of \textit{B. Chevreau}, \textit{G. Exner} and \textit{C. Pearcy} [Mich. Math. J. 36, 29-62 (1989; Zbl 0677.47001)]: given \(\varepsilon>0\) and \(f\in L^ 1(\Gamma)\), there exist vectors \(x,y\in H\) such that \(\| x\|\| y\|\leq(1+\varepsilon)\| f\|_ 1\) and \((p(T)x,y)=\int_ \Gamma pf\) for every polynomial \(p\) in one variable.