Jordan models and *-cyclic sets (Q1117436)

Let \(H^ 2({\mathfrak F})\) be the Hardy space of analytic functions f(\(\lambda)\) in the unit disc, with values vectors in a separable Hilbert space \({\mathfrak F}\). Let \(U_+\) be the shift operator \(f(\lambda)\to \lambda f(\lambda)\) and, for any given subset \({\mathfrak M}\) of \(H^ 2({\mathfrak F})\), form the subspace \[ {\mathfrak N}=\bigvee^{\infty}_{0}U^*_+ {\mathfrak M}\subset H^ 2({\mathfrak F}), \] invariant for \(U^*_+\). Then \(T=P_{{\mathfrak N}}U_+| {\mathfrak N}\) is a contraction operator on \({\mathfrak N}\). Suppose T is of class \(C_ 0\), i.e., \(\phi (T)=0\) for some non-zero \(\phi \in H^{\infty}\). The problem is to find the Jordan model of T directly from its *-cyclic set \({\mathfrak M}.\) \textit{A. E. Frazho} has considered the case of finite dimensional \({\mathfrak F}\) [Integral Equation Oper. Theory 5, 184-192 (1982; Zbl 0481.47005); Acta Sci. Math. (Szeged) 46, 317-321 (1983; Zbl 0579.47008)]. The present paper extends the study to the case of infinite-dimensional \({\mathfrak F}\), thereby using the ingenious method dealing with infinite-dimensional determinants, introduced in the important paper of \textit{H. Bercovici} and \textit{D. Voiculescu} ``Tensor operations on characteristic functions of \(C_ 0\) contractions'' [Acta Sci. Math. (Szeged) 39, 205-231 (1977; Zbl 0375.47006)].