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Zenon Jan Jabłoński

Publications and source records attributed to Zenon Jan Jabłoński.

17 recordsLinked to original sources
arXiv2026

Solutions to the 'moment property implies subnormality' and injectivity problems for composition operators on one-circuit graphs

We consider composition operators in $L^2(μ)$, where $μ$ is a discrete measure. Our main result shows that, for every integer $p\ge2$, there exists a non-hyponormal composition operator that generates Stieltjes moment sequences and whose symbol induces a directed graph consisting of a circuit of length $p$ and one infinite branch. In contrast, if the symbol induces a directed graph consisting of one loop and one infinite branch, then the generation of Stieltjes moment sequences does imply subnormality. These two results completely solve the ``moment property implies subnormality'' problem left open in [Adv. Math. 310 (2017), 484-556] for the entire family of one-branch directed graphs with a circuit of arbitrary length. Combining them with the examples in that paper shows that, for directed graphs consisting of one loop and a finite number of infinite branches, $2$ is the smallest number of branches for which the moment property does not imply subnormality. Finally, we construct a noninjective composition operator that generates Stieltjes moment sequences and whose symbol induces a directed graph consisting of one loop, one infinite branch, and one leaf. This operator gives a negative answer to the injectivity problem posed in the aforementioned paper. Both constructions use the Krein and Friedrichs measures of an arbitrary normalized S-indeterminate Stieltjes moment sequence.

math.FA↗
arXiv2026

CPD $n$th roots of subnormal operators are subnormal

We investigate the $n$th root problem for bounded operators on a Hilbert space within the class of conditionally positive definite (CPD) operators determined by the Lévy--Khintchine formula. The class contains subnormal operators, complete hypercontractions of order $2$, and $3$-isometries. Our main result shows that if $T$ is a CPD operator such that $T^n$ is subnormal (resp., quasinormal, normal, or a $3$-isometry), then $T$ belongs to the corresponding class. This establishes the invariance of these classes under taking $n$th roots within the CPD class and extends several earlier results in operator theory. Furthermore, we provide characterizations of quasinormal and normal operators in terms of their CPD property and the structure of the representing triplet. Finally, we show that the classes of CPD and normaloid operators are distinct by means of both theoretical arguments and explicit examples.

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arXiv2024

When is a CPD weighted shift similar to a subnormal operator?

We prove that a CPD unilateral weighted shift $W_λ$ of type III is a quasi-affine transform of the operator $M_z$ of multiplication by the independent variable on the $L^2(ρ)$-closure of analytic complex polynomials on the complex plane, where $ρ$ is a measure precisely determined by $W_λ$. By using this model, we provide necessary and sufficient conditions for similarity of $W_λ$ to $M_z$. Necessary conditions for a CPD operator to be similar to a subnormal one are given. A variety of concrete classes of non-subnormal CPD unilateral weighted shifts similar to subnormal operators are established.

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arXiv2024

Bishop-like theorems for non-subnormal operators

The celebrated Bishop theorem states that an operator is subnormal if and only if it is the strong limit of a net (or a sequence) of normal operators. By the Agler-Stankus theorem, $2$-isometries behave similarly to subnormal operator in the sense that the role of subnormal operators is played by $2$-isometries, while the role of normal operators is played by Brownian unitaries. In this paper we give Bishop-like theorems for $2$-isometries. Two methods are involved, the first of which goes back to Bishop's original idea and the second refers to Conway and Hadwin's result of general nature. We also investigate the strong and $*$-strong closedness of the class of Brownian unitaries.

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arXiv2024

Convergence of power sequences of operators via their stability

This paper is concerned with the convergence of power sequences and stability of Hilbert space operators, where "convergence" and "stability" refer to weak, strong and norm topologies. It is proved that an operator has a convergent power sequence if and only if it is a (not necessarily orthogonal) direct sum of an identity operator and a stable operator. This reduces the issue of convergence of the power sequence of an operator $T$ to the study of stability of $T$. The question of when the limit of the power sequence is an orthogonal projection is investigated. Among operators sharing this property are hyponormal and contractive ones. In particular, a hyponormal or a contractive operator with no identity part is stable if and only if its power sequence is convergent. In turn, a unitary operator has a weakly convergent power sequence if and only if its singular-continuous part is weakly stable and its singular-discrete part is the identity. Characterizations of the convergence of power sequences and stability of subnormal operators are given in terms of semispectral measures.

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arXiv2023

Convergence of power sequences of B-operators with applications to stability

The B-operators (abbreviation for Brownian-type operators) are upper triangular 2x2 block matrix operators that satisfy certain algebraic constraints. The purpose of this paper is to characterize the weak, the strongand the uniform stability of B-operators, respectively. This is achieved by giving equivalent conditions for the convergence of powers of a B-operator in each of the corresponding topologies. A more subtle characterization is obtained for B-operators with subnormal (2,2) entry. The issue of the strong stability of the adjoint of a B-operator is also discussed.

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arXiv2022

Conditionally positive definiteness in operator theory

In this paper we extensively investigate the class of conditionally positive definite operators, namely operators generating conditionally positive definite sequences. This class itself contains subnormal operators, $2$- and $3$-isometries and much more beyond them. Quite a large part of the paper is devoted to the study of conditionally positive definite sequences of exponential growth with emphasis put on finding criteria for their positive definiteness, where both notions are understood in the semigroup sense. As a consequence, we obtain semispectral and dilation type representations for conditionally positive definite operators. We also show that the class of conditionally positive definite operators is closed under the operation of taking powers. On the basis of Agler's hereditary functional calculus, we build an $L^{\infty}(M)$-functional calculus for operators of this class, where $M$ is an associated semispectral measure. We provide a variety of applications of this calculus to inequalities involving polynomials and analytic functions. In addition, we derive new necessary and sufficient conditions for a conditionally positive definite operator to be a subnormal contraction (including a telescopic one).

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arXiv2021

Conditionally positive definite unilateral weighted shifts

In a recent paper [15], Hilbert space operators $T$ with the property that each sequence of the form $\{\|T^n h\|^2\}_{n=0}^{\infty}$ is conditionally positive definite in a semigroup sense were introduced. In the present paper, this line of research is continued in depth in the case of unilateral weighted shifts. The conditional positive definiteness of weighted shifts is characterized in terms of formal moment sequences. The description of the representing triplet, the main object canonically associated with such operators, is provided. The backward extension problem for conditionally positive definite weighted shifts is solved, revealing a new feature that does not appear in the case of other operator classes. Finally, the flatness problem in this context is discussed, with an emphases on unexpected differences from the corresponding problem for subnormal weighted shifts.

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arXiv2020

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.

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arXiv2020

The Cauchy dual subnormality problem for cyclic $2$-isometries

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.

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arXiv2019

Taylor spectrum approach to Brownian-type operators with quasinormal entry

In this paper, we introduce operators that are represented by upper triangular $2\times 2$ block matrices whose entries satisfy some algebraic constraints. We call them Brownian-type operators of class $\mathcal Q,$ briefly operators of class $\mathcal Q.$ 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 $\mathcal Q$ 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 $\mathcal Q$ 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 $\mathcal Q.$ As a consequence, we solve the Cauchy dual subnormality problem for expansive operators of class $\mathcal Q$ 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 $\mathcal Q$ proving that the corresponding regions of subnormality are closed intervals with explicitly described endpoints.

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arXiv2018

Complete systems of unitary invariants for some classes of $2$-isometries

The unitary equivalence of $2$-isometric operators satisfying the so-called kernel condition is characterized. It relies on a model for such operators built on operator valued unilateral weighted shifts and on a characterization of the unitary equivalence of operator valued unilateral weighted shifts in a fairly general context. A complete system of unitary invariants for $2$-isometric weighted shifts on rooted directed trees satisfying the kernel condition is provided. It is formulated purely in the langauge of graph-theory, namely in terms of certain generation branching degrees. The membership of the Cauchy dual operators of $2$-isometries in classes $C_{0 \cdot}$ and $C_{\cdot 0}$ is also studied.

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arXiv2018

A solution to the Cauchy dual subnormality problem for 2-isometries

The Cauchy dual subnormality problem asks whether the Cauchy dual operator $T^{\prime}:=T(T^*T)^{-1}$ of a $2$-isometry $T$ is subnormal. In the present paper we show that the problem has a negative solution. The first counterexample depends heavily on a reconstruction theorem stating that if $T$ is a $2$-isometric weighted shift on a rooted directed tree with nonzero weights that satisfies the perturbed kernel condition, then $T^{\prime}$ is subnormal if and only if $T$ satisfies the (unperturbed) kernel condition. The second counterexample arises from a $2$-isometric adjacency operator of a locally finite rooted directed tree again by thorough investigations of positive solutions of the Cauchy dual subnormality problem in this context. We prove that if $T$ is a $2$-isometry satisfying the kernel condition or a quasi-Brownian isometry, then $T^{\prime}$ is subnormal. We construct a $2$-isometric adjacency operator $T$ of a rooted directed tree such that $T$ does not satisfy the kernel condition, $T$ is not a quasi-Brownian isometry and $T^{\prime}$ is subnormal.

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arXiv2012

Unbounded subnormal weighted shifts on directed trees I

A new method of verifying the subnormality of unbounded Hilbert space operators based on an approximation technique is proposed. Diverse sufficient conditions for subnormality of unbounded weighted shifts on directed trees are established. An approach to this issue via consistent systems of probability measures is invented. The role played by determinate Stieltjes moment sequences is elucidated. Lambert's characterization of subnormality of bounded operators is shown to be valid for unbounded weighted shifts on directed trees that have sufficiently many quasi-analytic vectors, which is a new phenomenon in this area.

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arXiv2012

Unbounded subnormal weighted shifts on directed trees. II

Criteria for subnormality of unbounded injective weighted shifts on leafless directed trees with one branching vertex are proposed. The case of classical weighted shifts is discussed. The relevance of an inductive limit approach to subnormality of weighted shifts on directed trees is revealed.

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arXiv2012

On Unbounded Composition Operators in $L^2$-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.

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