On the reflexivity of $C_{0}(N)$ contractions

Abstract:

Let T be a ${C_0}(N)$ contraction on a separable Hilbert space and let $J = S({\varphi _1}) \oplus S({\varphi _2}) \oplus \cdots \oplus S({\varphi _k})$ be its Jordan model, where ${\varphi _1},{\varphi _2}, \ldots ,{\varphi _k}$ are inner functions satisfying ${\varphi _j}|{\varphi _{j - 1}}$ for $j = 2,3, \ldots ,k$, and $S({\varphi _j})$ denotes the compression of the shift on ${H^2} \ominus {\varphi _j}{H^2},j = 1,2, \ldots ,k$. In this note we show that T is reflexive if and only if $S({\varphi _1}/{\varphi _2})$ is.