Extending an operator from a Hilbert space to a larger Hilbert space,so as to reduce its spectrum |
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Authors: | C J Read |
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Institution: | (1) Department of Pure Mathematics and Mathematical Statistics, Cambridge University, 16 Mill Lane, CB2 1SB Cambridge, England |
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Abstract: | In our earlier paper 1] we showed that given any elementx of a commutative unital Banach algebraA, there is an extensionA′ ofA such that the spectrum ofx inA′ is precisely the essential spectrum ofx inA. In 2], we showed further that ifT is a continuous linear operator on a Banach spaceX, then there is an extensionY ofX such thatT extends continuously to an operatorT
− onY, and the spectrum ofT
− is precisely the approximate point spectrum ofT. In this paper we take the second of these results, and show further that ifX is a Hilbert space then we can ensure thatY is also a Hilbert space; so any operatorT on a Hilbert spaceX is the restriction to one copy ofX of an operatorT
− onX ⊕X, whose spectrum is precisely the approximate point spectrum ofT. This result is “best possible” in the sense that if
isany extension to a larger Banach space of an operatorT, it is a standard exercise that the approximate point spectrum ofT is contained in the spectrum of
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Keywords: | |
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