A non-hyponormal operator generating Stieltjes moment sequences

Abstract

A linear operator S in a complex Hilbert space H for which the set D∞(S) of its C∞-vectors is dense in H and {‖Snf‖2}n=0∞ is a Stieltjes moment sequence for every f∈D∞(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, D∞(S) is a core of any power Sn 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 L2-space (over a σ-finite measure space) which is injective, paranormal and which generates Stieltjes moment sequences. The independence assertion of Barry Simonʼs theorem which parameterizes von Neumann extensions of a closed real symmetric operator with deficiency indices (1,1) is shown to be false.