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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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The inertial aspects of Stein’s condition $H-C^{\ast }\ HC\gg O$
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by Bryan E. Cain
Trans. Amer. Math. Soc. 196 (1974), 79-91
DOI: https://doi.org/10.1090/S0002-9947-1974-0350449-X

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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Bibliographic Information
  • © Copyright 1974 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 196 (1974), 79-91
  • MSC: Primary 47A10
  • DOI: https://doi.org/10.1090/S0002-9947-1974-0350449-X
  • MathSciNet review: 0350449