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Transactions of the American Mathematical Society

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

 

 

Subnormal operators, Toeplitz operators and spectral inclusion


Author: Gerard E. Keough
Journal: Trans. Amer. Math. Soc. 263 (1981), 125-135
MSC: Primary 47B20; Secondary 47B35
DOI: https://doi.org/10.1090/S0002-9947-1981-0590415-6
MathSciNet review: 590415
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Abstract: Let $ S$ be a subnormal operator on the Hilbert space $ H$, and let $ N = \int z \;dE(z)$ be its minimal normal extension on $ K$. Let $ \mu $ be a scalar spectral measure for $ N$. If $ f \in {L^\infty }(\mu )$, define $ {T_f} = Pf(N){\vert _H}:\;H \to H$, where $ P:K \to H$ denotes orthogonal projection. $ S$ has the $ {C^ \ast }$-Spectral Inclusion Property ( $ {C^ \ast }$-SIP) if $ \sigma (f(N)) \subseteq \sigma ({T_f})$, for all $ f \in C(\sigma (N))$, and $ S$ has the $ {W^\ast}$-Spectral Inclusion Property ($ {W^\ast}$-SIP) if $ \sigma (f(N)) \subseteq \sigma ({T_f})$, for all $ f \in {L^\infty }(\mu )$.

It is shown that $ S$ has the $ {C^\ast}$-SIP if and only if $ \sigma (N) = \Pi (S)$, the approximate point spectrum of $ S$. This is equivalent to requiring that $ E(\Delta )K$ have angle 0 with $ H$, for all nonempty, relatively open $ \Delta \subseteq \sigma (N)$. $ S$ has the $ {W^\ast}$-SIP if this angle condition holds for all proper Borel subsets of $ \sigma (N)$. If $ S$ is pure and has the $ {C^\ast}$ or $ {W^\ast}$-SIP, then it is shown that $ \sigma (f(N)) \subseteq {\sigma _e}({T_f})$, for all appropriate $ f$.


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DOI: https://doi.org/10.1090/S0002-9947-1981-0590415-6
Keywords: Subnormal operator, spectral inclusion theorems, angle between subspaces, reducing essential spectrum, extremely noncompact operator
Article copyright: © Copyright 1981 American Mathematical Society