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 \( \hat T \) 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\( \hat T \) .
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References
C. J. Read,Inverse producing extension of a commutative Banach algebra, which eliminates the residual spectrum of one element, Trans. Amer. Math. Soc.286 (1984).
C. J. Read,Inverse producing extension of a Banach space, which eliminates the residual spectrum of an operator, Trans. Amer. Math. Soc., to appear.
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Read, C.J. Extending an operator from a Hilbert space to a larger Hilbert space, so as to reduce its spectrum. Israel J. Math. 57, 375–380 (1987). https://doi.org/10.1007/BF02766221
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DOI: https://doi.org/10.1007/BF02766221