m--isometric composition operators on directed graphs with one circuit

The aim of this paper is to investigate $m$--isometric composition operators on directed graphs with one circuit. We establish a characterization of $m$--isometries and prove that complete hyperexpansiveness coincides with $2$--isometricity within this class. We discuss the $m$--isometric completion problem for unilateral weighted shifts and for composition operators on directed graphs with one circuit. The paper is concluded with an affirmative solution of the Cauchy dual subnormality problem in the subclass with circuit containing one element.Comment: 23 page

On the structure of conditionally positive definite algebraic operators

Recently, the authors have introduced and intensively studied a class of bounded Hilbert space operators called conditionally positive definite. Its origins go back to the harmonic analysis on ∗-semigroups, namely to the concept of conditional positive definiteness. Our main aim here is to give a complete description of algebraic condi tionally positive definite operators on inner product spaces; we do not assume that the operators under consideration are bounded

The Cauchy dual subnormality problem for cyclic 2-isometrie

The Cauchy dual subnormality problem asks whether the Cauchy dual operator of a 2-isometry is subnormal. Recently this problem has been solved in the negative. Here we show that it has a negative solution even in the class of cyclic 2-isometries

Taylor spectrum approach to Brownian-type operators with quasinormal entry

In this paper, we introduce operators that are represented by upper triangular 2×2 block matrices whose entries satisfy some algebraic constraints. We call them Brownian-type operators of class Q, briefly operators of class Q. These operators emerged from the study of Brownian isometries performed by Agler and Stankus via detailed analysis of the time shift operator of the modified Brownian motion process. It turns out that the class Q is closely related to the Cauchy dual subnormality problem which asks whether the Cauchy dual of a completely hyperexpansive operator is subnormal. Since the class Q is closed under the operation of taking the Cauchy dual, the problem itself becomes a part of a more general question of investigating subnormality in this class. This issue, along with the analysis of nonstandard moment problems, covers a large part of the paper. Using the Taylor spectrum technique culminates in a full characterization of subnormal operators of class Q. As a consequence, we solve the Cauchy dual subnormality problem for expansive operators of class Q in the affirmative, showing that the original problem can surprisingly be extended to a class of operators that are far from being completely hyperexpansive. The Taylor spectrum approach turns out to be fruitful enough to allow us to characterize other classes of operators including m-isometries. We also study linear operator pencils associated with operators of class Q proving that the corresponding regions of subnormality are closed intervals with explicitly described endpoints

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