Characterization of the unbounded bicommutant of C_0 (N) contractions

Abstract

Recent results have shown that any closed operator $A$ commuting with the backwards shift $S^*$ restricted to $K ^2_u := H^2 \ominus u H^2$, where $u$ is an inner function, can be realized as a Nevanlinna function of $S^*_u := S^* |_{K^2_u}$, $A = \varphi (S^*_u)$, where $\varphi$ belongs to a certain class of Nevanlinna functions which depend on $u$. In this paper this result is generalized to show that given any contraction $T$ of class $C_0 (N)$, that any closed (and not necessarily bounded) operator $A$ commuting with the commutant of $T$ is equal to $\varphi (T)$ where $\varphi $ belongs to a certain class of Nevanlinna functions which depend on the minimal inner function $m_T$ of $T$.