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 $S^{\ast }$ restricted to $K_u^2:=H^2\ominus u{H}^2$, where $u$ is an inner function, can be realized as a Nevanlinna function of $S_u^{\ast }:=S^{\ast }{|}_{K_u^2}$, $A=\phi (S_u^{\ast })$, where $\phi$ 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 $\phi (T)$ where $\phi$ belongs to a certain class of Nevanlinna functions which depend on the minimal inner function $m_T$ of $T$.

accepted for publication by Operators and Matrices on Aug. 7, 2009. See arXiv:0908.2262 by H. Bercovici for related and extended results

Cited by in corpus (1)

• The unbounded commutant of an operator of class C_0