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

MathSciNet review:
590415

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a subnormal operator on the Hilbert space , and let be its minimal normal extension on . Let be a scalar spectral measure for . If , define , where denotes orthogonal projection. has the -Spectral Inclusion Property ( -SIP) if , for all , and has the -Spectral Inclusion Property (-SIP) if , for all .

It is shown that has the -SIP if and only if , the approximate point spectrum of . This is equivalent to requiring that have angle 0 with , for all nonempty, relatively open . has the -SIP if this angle condition holds for all proper Borel subsets of . If is pure and has the or -SIP, then it is shown that , for all appropriate .

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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