Unbounded subnormal composition operators in L2-spaces

Abstract

A criterion for subnormality of unbounded composition operators in L 2 -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 ∞ -vectors in the underlying ℓ 2 -space.2010 Mathematics Subject Classification. Primary 47B33, 47B20; Secondary 47B37, 44A60.

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

“…In this section we concentrate on a family of weighted composition operators coming from [7,Example 42] (see also [8,Section 3.2]). Let w : Z + → C be any function.…”

Operators whose ranges are contained in the null space of conditional expectations

“…The densely defined composition operator induced by φ is given by , for each . Here, the non‐singularity of φ guarantees that is well defined and closed (see ). A recent monograph related to unbounded composition operators is .…”

“…Here, the non‐singularity of φ guarantees that is well defined and closed (see ). A recent monograph related to unbounded composition operators is . Let .…”

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.…”