Unbounded subnormal weighted shifts on directed trees

Abstract

Abstract. 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.

A Subnormal Completion Problem for Weighted Shifts on Directed Trees, II

“…The class of unilateral weighted shifts on Hilbert space has been a standard and important source of examples with which to study the properties of bounded linear operators on Hilbert space, including especially the investigation of subnormality (see [18] and [5]). A recently introduced class of weighted shifts on directed trees provides a more extensive collection of objects for study (see e.g., [15,14,1,2,3]). In [10], we initiated the study of a subnormal completion problem for weighted shift operators on directed trees.…”

Multishifts on directed Cartesian products of rooted directed trees

“…The investigations in the present work are related to the idea of shifts associated with discrete structures (e.g. directed trees), recently boosted in the theory of Hilbert space operators ( [65], [25], [66], [26], [67], [68], [27], [28], [29], [33], [79]). The significantly large class of weighted shift operators on directed trees contains all classical weighted shifts and has an overlap with that of composition operators.…”

Quasinormal extensions of subnormal operator-weighted composition operators in ℓ2-spaces

“…Recent years have brought rapid development in studies over unbounded composition operators in L 2 -spaces (see [5,6,7,8,11,12,13,15,19,28,34]) and weighted shifts on directed trees (see [9,10,14,29,30,31,33,41,49]), mostly in connection with the question of their subnormality. One can see that all these operators belong to a larger class of Hilbert space operators composed of unbounded weighted composition operators in L 2 -spaces (see [16]).…”