The exceptional subset of a $C_{0}$-contraction

Abstract:

Let $T$ be a ${C_0}$-operator acting on a (complex separable) Hilbert space $\mathcal {K}$; i.e., $T$ is a contraction on $\mathcal {K}$ and it satisfies the equation $q(T) = 0$ for some inner function $q$, where $q(T)$ is defined in the sense of the functional calculus of B.Sz.-Nagy and C. Foiaş. Among all those inner functions $q$ there exists a unique minimal function $p$ defined by the conditions: (1) $p(T) = 0$; (2) if $q(T) = 0$, then $p$ divides $q$. A vector $F \in \mathcal {K}$ is called exceptional if there exists an inner function $r$ such that $r(T)F = 0$, but $p$ does not divide $r$. The existence of nonexceptional vectors plays a very important role in the theory of ${C_0}$-operators. The main result of this paper says that nonexceptional vectors actually exist; moreover, the exceptional subset of a ${C_0}$-operator is a topologically small subset of $\mathcal {K}$.