Jan Stochel

Seminormality of operators from their tensor product

The question of seminormality of tensor products of nonzero bounded linear operators on Hilbert spaces is investigated. It is shown that A 09 B is subnormal if and only if so are A and B.

Solving the truncated moment problem solves the full moment problem

It is shown that the truncated multidimensional moment problem is more general than the full multidimensional moment problem.

A hyponormal weighted shift on a directed tree whose square has trivial domain

It is proved that, up to isomorphism, there are only two directed trees that admit a hyponormal weighted shift with nonzero weights whose square has trivial domain. These are precisely those enumerable directed trees, one with root, the other without, whose every vertex has enumerable set of successors.

An asymmetric Putnam-Fuglede theorem for unbounded operators

The intertwining relations between cosubnormal and closed hyponormal (resp. cohyponormal and closed subnormal) operators are studied. In particular, an asymmetric Putnam–Fuglede theorem for unbounded operators is proved.

On unbounded composition operators in L^2-spaces

Fundamental properties of unbounded composition operators in

UNBOUNDED 2-HYPEREXPANSIVE OPERATORS

L^2

Lifting strong commutants of unbounded subnormal operators

-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.

A peculiarity of the creation operator

A symmetric operator X^ is attached to each operator X that leaves the domain of a given positive operator A invariant and makes the product AX symmetric. Some spectral properties of X^ are derived from those of X and, as a consequence, various conditions ensuring positivity of products of the form AX1 ... Xn are proved. The question of ^-complete positivity of the mapping p → Ap(X1,...,Xn) defined on complex polynomials in n variables is investigated. It is shown that the set ω is related to the McIntosh-Pryde joint spectrum of (X1,...,Xn) in case all the operators A, X1,...,Xn are bounded. Examples illustrating the theme of the paper are included.