Bounded characteristic functions and models for noncontractive sequences of operators. (Q1865895)

In the paper under review, the Sz.-Nagy-Foiaş functional model for completely non-unitary contractions is extended to completely non-coisometric sequences of bounded operators on a Hilbert space \(H\). It was shown in [\textit{B. Sz.-Nagy} and \textit{C. Foiaş}, C. R. Acad. Sci., Paris 256, 3236--3238 (1963; Zbl 0151.19504)] that every completely non-unitary contraction on a Hilbert space is unitarily equivalent to a multiplication operator on a Hilbert space of operator valued functions. This paper induced an enormous body of research in the last decades. In their previous works (e.g., [17,19] in the references) the authors extended these results to sequences of operators \((T_i)\) such that the matrix \([T_1,T_2,\dots]\) is a contraction. It was also demonstrated in these works that there is a natural connection to multi-analytic operators on a Fock space with indefinite inner product. The present paper continues these investigations. The main result (Theorem 4.1) gives necessary and sufficient conditions for a bounded multi-analytic operator on a Fock space to coincide with the characteristic function associated to a completely non-coisometric sequence of bounded operators on a Hilbert space.