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A non-hyponormal operator generating Stieltjes moment sequences. (English) Zbl 1258.47026
One says that a linear operator $S$ acting in a complex Hilbert space $\mathcal H$ generates Stieltjes moment sequences if the set $\mathcal D^\infty(S)$ of its $C^\infty$-vectors is dense in $\mathcal H$ and $\{\Vert S^nf\Vert^2\}_{n=0}^\infty$ is a Stieltjes moment sequence for every $f\in\mathcal D^\infty(S)$. The aim of the paper is to show that there exists a closed non-hyponormal operator $S$ which generates Stieltjes moment sequences. In particular, there exists a non-hyponormal composition operator in an $L^2$-space which is injective, paranormal and which generates Stieltjes moment sequences. The authors also show that the independence assertion of {\it B. Simon}’s theorem (see Theorem A.1.1 in [Adv. Math. 137, No. 1, 82--203 (1998; Zbl 0910.44004)]) which parametrize von Neumann extensions of a closed real symmetric operator with indices (1,1) is false.

MSC:
47A57Operator methods in interpolation, moment and extension problems
47B20Subnormal operators, hyponormal operators, etc.
47B25Symmetric and selfadjoint operators (unbounded)
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References:
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