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Subnormality and generalized commutation relations. (English) Zbl 0656.47015

Let S be a linear operator defined on a dense subset D(S) of a Hilbert space H, and let \(D(S^*)\) denote the domain of its adjoint. Let \(M\subset D(S)\cap D(S^*)\) be a dense subset of H such that SM\(\subset M\) and \(S^*M\subset M\). Assume there exists a (not necessarily bounded) operator E such that (1) \(M\subset D(E)\cap D(E^*)\), (2) EM\(\subset M\), (3) \((S^*S-SS^*)f=E^ 2f\) for all \(f\in M\), (4) \(SEf=ESf\) for all \(f\in M\), (5) \((f,Eg)=(Ef,g)\) for all \(f,g\in M.\)
The main result of the paper is to show that such an operator S satisfies the Halmos-Bram condition on M; that is, \(\sum^{n}_{i,j=0}(S^ jf_ i,S^ if_ j)\geq 0\) for all finite collection \(\{f_ 0,f_ 1,...,f_ n\}\subset M\) and all natural numbers n.
Reviewer: M.Radjabalipour

MSC:

47B20 Subnormal operators, hyponormal operators, etc.
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References:

[1] Stochel, On normal extension of unbounded operators II (1985) · Zbl 0613.47022
[2] Stochel, J. Operator. Theory 14 pp 31– (1985)
[3] Putnam, Commutation properties of Hilbert space operators and related topics (1967) · Zbl 0149.35104 · doi:10.1007/978-3-642-85938-0
[4] DOI: 10.1002/cpa.3160140303 · Zbl 0107.09102 · doi:10.1002/cpa.3160140303
[5] Halmos, A Hilbert space problem book (1967)
[6] Conway, Subnormal operators (1987)
[7] Coddington, Canad. J. Math. 17 pp 1030– (1965) · Zbl 0134.31803 · doi:10.4153/CJM-1965-098-8
[8] DOI: 10.1016/0022-247X(87)90327-1 · Zbl 0626.47042 · doi:10.1016/0022-247X(87)90327-1
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