A non-hyponormal operator generating Stieltjes moment sequences

TL;DR

This paper constructs a non-hyponormal operator in a Hilbert space that generates Stieltjes moment sequences, revealing new properties of such operators and their relation to composition operators and extension theory.

Contribution

It demonstrates the existence of a closed non-hyponormal operator generating Stieltjes moment sequences using weighted shifts on directed trees, and explores implications for composition operators and extension theory.

Findings

Existence of a non-hyponormal operator generating Stieltjes moment sequences.

Construction of such an operator via weighted shift on a directed tree.

Counterexample to the independence assertion of Barry Simon's theorem.

Abstract

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.