A non-hyponormal operator generating Stieltjes moment sequences
By Zenon Jan Jablo'nski, IL Bong Jung, Jan Stochel. 2011
48 citations recorded by the source corpus.
A linear operator $S$ in a complex Hilbert space $\hh$ for which the set $\dzn{S}$ of its $C^\infty$-vectors is dense in $\hh$ and $\{\|S^n f\|^2\}_{n=0}^\infty$ is a Stieltjes moment sequence for every $f \in \dzn{S}$ is said to generate Stieltjes moment sequences. It is shown that there exists a closed non-hyponormal operator $S$ which generates Stieltjes moment sequences. What is more, $\dzn{S}$ is a core of any power $S^n$ of $S$. This is established with the help of a weighted shift on a directed tree with one branching vertex. The main tool in the construction comes from the theory of indeterminate Stieltjes moment sequences. As a consequence, it is shown that there exists a non-hyponormal composition operator in an $L^2$-space (over a $\sigma$-finite measure space) which is injective, paranormal and which generates Stieltjes moment sequences. In contrast to the case of abstract Hilbert space operators, composition operators which are formally normal and which generate Stieltjes moment sequences are always subnormal (in fact normal). The independence assertion of Barry Simon's theorem which parameterizes von Neumann extensions of a closed real symmetric operator with deficiency indices $(1,1)$ is shown to be false.
Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/bb789e205e28bf4811c55f6fbf54f37eda12756a
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In Applied Mathematics
In Mathematical Physics
Citations in the corpus
- A solution to the Cauchy dual subnormality problem for 2-isometries (2017, background): “ collect selected properties of weighted shifts on directed trees that we need for further considerations. This class of operators was introduced in [28] and intensively studied since then (see e.g., [29, 23, 14, 9, 11, 33, 30]). We show, among other things, that the Cauchy dual of a left-invertible”