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

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 $\beta (C) < \infty$.