A non-hyponormal operator generating Stieltjes moment sequences

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

On Unbounded Composition Operators in L2L^2-Spaces

Fundamental properties of unbounded composition operators in $L^2$-spaces are studied. Characterizations of normal and quasinormal composition operators are provided. Formally normal composition operators are shown to be normal. Composition operators generating Stieltjes moment sequences are completely characterized. The unbounded counterparts of the celebrated Lambert's characterizations of subnormality of bounded composition operators are shown to be false. Various illustrative examples are supplied

Characterizations of positive selfadjoint extensions

The set of all positive selfadjoint extensions of a positive operator T (which is not assumed to be densely defined) is described with the help of the partial order which is relevant to the theory of quadratic forms. This enables us to improve and extend a result of M. G. Krein to the case of not necessarily densely defined operators T. © 2006 American Mathematical Society

Unbounded Toeplitz Operators in the Segal-Bargmann Space, II

AbstractThe paper mostly deals with the questions of closedness and essential selfadjointness of Toeplitz operators in the Segal-Bargmann space. General criteria for their closedness are formulated. Examples of unclosed Toeplitz operators are given. Explicit formulas for their adjoints are shown for various classes of symbols. The problem of whether polynomials and exponents form cores for Toeplitz operators is investigated. The results presented here improve and extend the ones from an earlier paper