Characterization of the unbounded bicommutant of C_0 (N) contractions

TL;DR

This paper extends the characterization of operators commuting with certain contractions, showing they can be represented as Nevanlinna functions of the contraction, generalizing previous results for specific shift operators.

Contribution

It generalizes existing results by characterizing unbounded operators commuting with the commutant of any contraction of class C_0(N) as Nevanlinna functions of that contraction.

Findings

Operators commuting with the commutant are representable as Nevanlinna functions.

The result applies to all contractions of class C_0(N), not just specific shift operators.

The characterization depends on the minimal inner function of the contraction.

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 $T$.