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.

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.

Various formulas for reciprocals of densely defined weighted composition operators in L 2 -spaces as well as for their adjoints are provided. The relation between the reciprocal of a weighted composition operator and the product of the reciprocals of the associated multiplication operator and composition operator are studied.

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. 2010 Mathematics Subject Classification. Primary 47B33, 47B20; Secondary 47A05. Key words and phrases. Composition operator in L 2 -space, normal operator, quasinormal operator, formally normal operator, subnormal operator, operator generating Stieltjes moment sequences. 1 2 P. BUDZYŃSKI, Z. J. JAB LOŃSKI, I. B. JUNG, AND J. STOCHELcharacterizations if and only if it generates Stieltjes moment sequences (see Definition 2.3 and Theorem 10.4). Thus, knowing that there exists a non-subnormal composition operator which generates Stieltjes moment sequences (see [25, Theorem 4.3.3]), we obtain the above-mentioned result (see Conclusion 10.5). We point out that there exists a non-subnormal formally normal operator which generates Stieltjes moment sequences (for details see [7, Section 3.2]). This is never the case for composition operators because, as shown in Theorem 9.4, each formally normal composition operator is normal, and as such subnormal. We refer the reader to [48,49,50,51] for the foundations of the theory of unbounded subnormal operators (for the bounded case see [21,14]).The above discussion makes plain the importance of the question of when C ∞vectors of a composition operator form a dense subset of the underlying L 2 -space. This and related topics are studied in Section 4. In Section 3, we collect some necessary facts on composition operators. Illustrative examples are gathered in Section 5. In Section 6, we address the question of injectivity of composition operators. In Section 7, we describe the polar decomposition of a composition operator. Next, in Sections 8 and 9, we characterize normal, quasinormal and formally normal composition operators. Finally, in Section 10, we investigate composition operators which generate Stieltjes moment sequences. We conclude the paper with two appendices. In Appendix A we gather particular properties of L 2 -spaces exploited throughout the paper. Appendix B is mostly devoted to the operator of conditional expectation which plays an essential role in our investigations.Caution. All measure spaces being considered in this paper, except for Appendices A and B, are assumed to be σ-finite. PreliminariesDenote by C, R and R + the sets of complex numbers, real numbers and nonnegative real numbers, respectively. We write Z + for the set of all nonnegative integers, and N for the set of all positive integers. The characteristic function of a subset ∆ of a set X will be denoted by χ ∆ . We write ∆ △ ∆ ′ = (∆ \ ∆ ′ ) ∪ (∆ ′ \ ∆) for subsets ∆ and ∆ ′ of X. Given a sequence {∆ n } ...

We show that a weighted shift on a directed tree is related to a multiplier algebra of coefficients of analytic functions. We use this relation to study spectral properties of the operator in question.