On the spectra of contractions belonging to special classes (Q1073311)

Let T be a contraction acting on the Hilbert space H. Let us assume that T is of class \(C_{10}\) that is \(\lim_{n\to \infty}\| T^ nh\| \neq 0=\lim_{n\to \infty}\| T^{*n}h\|\), for every nonzero vector h. Then a new scalar product can be introduced in H by \(<h,k>_{\sim}=\lim_{n\to \infty}<T^ nh,T^ nk>\). T acts as an isometry on the inner product space \((H,<\cdot,\cdot >_{\sim})\). Let \(\tilde T\) denote the minimal unitary extension of this isometry acting on the Hilbert space \(\tilde H.\) The main result of the paper is a complete characterization of the possible spectra of a \(C_{10}\)- contraction T and its minimal unitary extension \(\tilde T.\) The theorem is analogous to the one obtained for \(C_{11}\)-contractions in the work [Proc. Amer. Math. Soc. 95, 412-418 (1985)], jointly with \textit{H. Bercovici}; however the proof is more difficult.