Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 $.

References [Enhancements On Off] (What's this?)

  • [1] Bryan E. Cain, An inertia theory for operators on a Hilbert space, J. Math. Anal. Appl. 41 (1973), 97–114. MR 0317089
  • [2] Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. MR 0117523
  • 1. -, Linear operators. II: Spectral theory. Selfadjoint operators in Hilbert space, Interscience, New York, 1963. MR 32 #6181.
  • [3] Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
  • [4] Alexander Ostrowski and Hans Schneider, Some theorems on the inertia of general matrices, J. Math. Anal. Appl. 4 (1962), 72–84. MR 0142555
  • [5] Frigyes Riesz and Béla Sz.-Nagy, Functional analysis, Frederick Ungar Publishing Co., New York, 1955. Translated by Leo F. Boron. MR 0071727
  • [6] Olga Taussky, A generalization of a theorem of Lyapunov, J. Soc. Indust. Appl. Math. 9 (1961), 640–643. MR 0133336
  • [7] Olga Taussky, Matrices 𝐶 with 𝐶ⁿ→0, J. Algebra 1 (1964), 5–10. MR 0161865
  • [8] James P. Williams, Similarity and the numerical range, J. Math. Anal. Appl. 26 (1969), 307–314. MR 0240664

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 47A10

Retrieve articles in all journals with MSC: 47A10

Additional Information

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