A non-hyponormal operator generating Stieltjes moment sequences

作者：Jablonski Zenon Jan; Jung Il Bong^*; Stochel Jan

来源：Journal of Functional Analysis, 2012, 262(9): 3946-3980.

DOI：10.1016/j.jfa.2012.02.006

• Indeterminate moment problem
• N-extremal measure
• Krein and Friedrichs measures
• Directed tree
• Weighted shift on a directed tree
• Hyponormal operator
• Operator generating Stieltjes moment sequences
• Composition operator in an L-2-space

A linear operator S in a complex Hilbert space H for which the set D-infinity (S) of its C-infinity-vectors is dense in H and {parallel to S-n f parallel to(2)}(n=0)(infinity) is a Stieltjes moment sequence for every f is an element of D-infinity (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-infinity (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. The independence assertion of Barry Simon%26apos;s theorem which parameterizes von Neumann extensions of a closed real symmetric operator with deficiency indices (1, 1) is shown to be false.

• 出版日期2012-5-1