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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The exceptional subset of a $ C\sb{0}$-contraction

Author: Domingo A. Herrero
Journal: Trans. Amer. Math. Soc. 173 (1972), 93-115
MSC: Primary 47A45; Secondary 47A15
MathSciNet review: 0310679
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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}$.

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Keywords: Sz.-Nagy and Foiaş' functional calculus, Baire category theorems, $ {C_{ \cdot 0}}$-, $ {C_{00}}$-, $ {C_0}$-contractions, $ {C_{00}}$-weak contractions, invariant subspace, full-range invariant subspace, IN-subspace, determinant class, inner function-operator, IN-operator, minimal inner function, characteristic function, analytic direction, exceptional subset, quasi-similar transformation, Jordan model
Article copyright: © Copyright 1972 American Mathematical Society