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Transactions of the American Mathematical Society

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



The inertial aspects of Stein's condition $ H-C\sp{\ast}\ HC\gg O$

Author: Bryan E. Cain
Journal: Trans. Amer. Math. Soc. 196 (1974), 79-91
MSC: Primary 47A10
MathSciNet review: 0350449
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Abstract: To each bounded operator C on the complex Hilbert space $ \mathcal{H}$ we associate the vector space $ {\mathcal{K}_C}$ consisting of those $ x \in \mathcal{H}$ for which $ {C^n}x \to 0$ as $ n \to \infty $. We let $ \alpha (C)$ denote the dimension of the closure of $ {\mathcal{K}_C}$ and we set $ \beta (C) = \dim (\mathcal{K}_C^ \bot )$. Our main theorem states that if H is Hermitian and if $ H - {C^ \ast }HC$ is positive and invertible then $ \alpha (C) \leq \pi (H),\beta (C) = \nu (H)$, and $ \beta (C) \geq \delta (H)$ where $ (\pi (H),\nu (H),\delta (H))$ is the inertia of H. (That is, $ \pi (H) = \dim \;({\text{Range}}\;E[(0,\infty )])$) where E is the spectral measure of H; $ \nu (H) = \pi ( - H)$; and $ \delta (H) = \dim ({\operatorname{Ker}}\;H)$.) We also show (l) that in general no stronger conclusion is possible, (2) that, unlike previous inertia theorems, our theorem allows 1 to lie in $ \sigma (C)$, the spectrum of C, and (3) that the main inertial results associated with the hypothesis that $ \operatorname{Re} (HA)$ is positive and invertible can be derived from our theorem. Our theorems (1) characterize C in the extreme cases that either $ \pi (H) = 0$ or $ \nu (H) = 0$, and (2) prove that $ \alpha (C) = \pi (H),\beta (C) = \nu (H),\delta (H) = 0$ if either $ 1 \notin \sigma (C)$ or $ \beta (C) < \infty $.

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Keywords: Direct sum decomposition, Hilbert space, inertia, invariant subspace, Lyapunov, spectral measure, spectrum, Stein
Article copyright: © Copyright 1974 American Mathematical Society