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Il Bong Jung

Publications and source records attributed to Il Bong Jung.

27 records · Page 2Linked to original sources
arXiv2013

Unbounded Subnormal Composition Operators in L2-Spaces

A criterion for subnormality of unbounded composition operators in L2-spaces, written in terms of measurable families of probability measures satisfying the so-called consistency condition, is established. It becomes a new characterization of subnormality in the case of bounded composition operators. Pseudo-moments of a measurable family of probability measures that satisfies the consistency condition are proved to be given by the Radon-Nikodym derivatives which appear in Lambert's characterization of bounded composition operators. A criterion for subnormality of composition operators induced by matrices is provided. The question of subnormality of composition operators over discrete measure spaces is studied. Two new classes of subnormal composition operators over discrete measure spaces are introduced. A recent criterion for subnormality of weighted shifts on directed trees by the present authors is essentially improved in the case of rootless directed trees and nonzero weights by dropping the assumption of density of C\infty-vectors in the underlying L2-space.

math.FA↗
arXiv2012

Operators with absolute continuity properties: an application to quasinormality

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 emerged in this context are invented. Various examples and counterexamples illustrating the concepts of the paper are constructed by means of weighted shifts on directed trees. Generalizations of these results that cover the case of q-quasinormal operators are established.

math.FA↗
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.

math.FA↗
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.

math.FA↗
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.

math.FA↗
arXiv2011

Unbounded subnormal weighted shifts on directed trees

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. The cases of classical weighted shifts and weighted shifts on leafless directed trees with one branching vertex are studied.

math.FA↗
arXiv2009

Weighted shifts on directed trees

A new class of (not necessarily bounded) operators related to (mainly infinite) directed trees is introduced and investigated. Operators in question are to be considered as a generalization of classical weighted shifts, on the one hand, and of weighted adjacency operators, on the other; they are called weighted shifts on directed trees. The basic properties of such operators, including closedness, adjoints, polar decomposition and moduli are studied. Circularity and the Fredholmness of weighted shifts on directed trees are discussed. The relationships between domains of a weighted shift on a directed tree and its adjoint are described. Hyponormality, cohyponormality, subnormality and complete hyperexpansivity of such operators are entirely characterized in terms of their weights. Related questions that arose during the study of the topic are solved as well. Particular trees with one branching vertex are intensively studied mostly in the context of subnormality and complete hyperexpansivity of weighted shifts on them. A strict connection of the latter with $k$-step backward extendibility of subnormal as well as completely hyperexpansive unilateral classical weighted shifts is established. Models of subnormal and completely hyperexpansive weighted shifts on these particular trees are constructed. Various illustrative examples of weighted shifts on directed trees with the prescribed properties are furnished. Many of them are simpler than those previously found on occasion of investigating analogical properties of other classes of operators.

math.FA↗