### Algebraic operators and moments on algebraic sets.

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The paper deals with operator-valued positive definite kernels on a convex *-semigroup whose Kolmogorov-Aronszajn type factorizations induce *-semigroups of bounded shift operators. Any such kernel Φ has a canonical decomposition into a degenerate and a nondegenerate part. In case is commutative, Φ can be disintegrated with respect to some tight positive operator-valued measure defined on the characters of if and only if Φ is nondegenerate. It is proved that a representing measure of a positive...

Isometric automorphisms of normed linear spaces are characterized by suitable concavity properties of powers of operators. Bounded selfadjoint operators in Hilbert spaces are described by parallel concavity properties of the exponential group. Unbounded infinitesimal generators of 𝓒₀-groups of Hilbert space operators having concavity properties are characterized as well.

A generalization of the Carleman criterion for selfadjointness of Jacobi matrices to the case of symmetric matrices with finite rows is established. In particular, a new proof of the Carleman criterion is found. An extension of Jørgensen's criterion for selfadjointness of symmetric operators with "almost invariant" subspaces is obtained. Some applications to hyponormal weighted shifts are given.

Joint subnormality of a family of composition operators on L²-space is characterized by means of positive definiteness of appropriate Radon-Nikodym derivatives. Next, simplified positive definiteness conditions guaranteeing joint subnormality of a C₀-semigroup of composition operators are supplied. Finally, the Radon-Nikodym derivatives associated to a jointly subnormal C₀-semigroup of composition operators are shown to be the Laplace transforms of probability measures (modulo a C₀-group of scalars)...

In the previous paper, we have characterized (joint) subnormality of a C₀-semigroup of composition operators on L²-space by positive definiteness of the Radon-Nikodym derivatives attached to it at each rational point. In the present paper, we show that in the case of C₀-groups of composition operators on L²-space the positive definiteness requirement can be replaced by a kind of consistency condition which seems to be simpler to work with. It turns out that the consistency condition also characterizes...

The following two questions as well as their relationship are studied: (i) Is a closed linear operator in a Banach space bounded if its ${}^{\infty}$-vectors coincide with analytic (or semianalytic) ones? (ii) When are the domains of two successive powers of the operator in question equal? The affirmative answer to the first question is established in case of paranormal operators. All these investigations are illustrated in the context of weighted shifts.

The main goal of this paper is to clarify the antisymmetric nature of a binary relation ≪ which is defined for normal operators A and B by: A ≪ B if there exists an operator T such that ${E}_{A}\left(\Delta \right)\le T*{E}_{B}\left(\Delta \right)T$ for all Borel subset Δ of the complex plane ℂ, where ${E}_{A}$ and ${E}_{B}$ are spectral measures of A and B, respectively (the operators A and B are allowed to act in different complex Hilbert spaces). It is proved that if A ≪ B and B ≪ A, then A and B are unitarily equivalent, which shows that the relation ≪ is a partial order...

The concept of k-step full backward extension for subnormal operators is adapted to the context of completely hyperexpansive operators. The question of existence of k-step full backward extension is solved within this class of operators with the help of an operator version of the Levy-Khinchin formula. Some new phenomena in comparison with subnormal operators are found and related classes of operators are discussed as well.

An absolute continuity approach to quasinormality which relates the operator in question to the spectral measure of its modulus is developed. Algebraic characterizations of some classes of operators that emerge in this context are found. Various examples and counterexamples illustrating the concepts of the paper are constructed by using weighted shifts on directed trees. Generalizations of these results that cover the case of q-quasinormal operators are established.

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