1 Introduction

Composition operators (in \(L^2\)-spaces over \(\sigma \)-finite spaces), which play an essential role in Ergodic Theory, turn out to be interesting objects of Operator Theory. The questions of boundedness, normality, quasinormality, subnormality, hyponormality etc. of such operators have been answered (cf. [810, 16, 17, 19, 20, 23, 3032, 34, 4244, 54, 56]; see also [15, 18, 33, 45, 47] for particular classes of composition operators). This means that the theory of bounded composition operators on \(L^2\)-spaces is well-developed.

The literature on unbounded composition operators in \(L^2\)-spaces is meager. So far, only the questions of seminormality, k-expansivity, and complete hyperexpansivity have been studied (cf. [11, 24]). Very little is known about other properties of unbounded composition operators. To the best of our knowledge, there is no paper concerning the issue of subnormality of such operators. It is a difficult question mainly because Lambert’s criterion for subnormality of bounded operators (cf. [29]) is no longer valid for unbounded ones. In the present paper, we show that the unbounded counterparts of the celebrated Lambert’s characterizations of subnormality of bounded composition operators given in [31] fail to hold. This is achieved by proving that a composition operator satisfies the requirements of Lambert’s characterizations if and only if it generates Stieltjes moment sequences (see Definition 2.3, 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 Sect. 11). 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 [4851] for the foundations of the theory of unbounded subnormal operators (for the bounded case see [14, 21]).

The above discussion makes plain the importance of the question of when \(C^\infty \)-vectors of a composition operator form a dense subset of the underlying \(L^2\)-space. This and related topics are studied in Sect. 4. In Sect. 3, we collect some necessary facts on composition operators. Illustrative examples are gathered in Sect. 5. In Sect. 6, we address the question of injectivity of composition operators. In Sect. 7, we describe the polar decomposition of a composition operator. Next, in Sects. 8 and 9, we characterize normal, quasinormal and formally normal composition operators. Finally, in Sect. 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 \(\sigma \)-finite.

2 Preliminaries

Denote by \(\mathbb{C }, \mathbb{R }\) and \(\mathbb{R }_+\) the sets of complex numbers, real numbers and nonnegative real numbers, respectively. We write \(\mathbb{Z }_+\) for the set of all nonnegative integers and \(\mathbb N \) for the set of all positive integers. The characteristic function of a subset \(\varDelta \) of a set \(X\) will be denoted by \(\chi _\varDelta \). We write \(\varDelta \vartriangle \varDelta ^\prime \!=\! (\varDelta {\setminus } \varDelta ^\prime ) \cup (\varDelta ^\prime \setminus \varDelta )\) for subsets \(\varDelta \) and \(\varDelta ^\prime \) of \(X\). Given a sequence \(\{\varDelta _n\}_{n=1}^\infty \) of subsets of \(X\) and a subset \(\varDelta \) of \(X\) such that \(\varDelta _n \subseteq \varDelta _{n\!+\!1}\) for every \(n\in \mathbb N \), and \(\varDelta \!=\!\bigcup _{n\!=\!1}^\infty \varDelta _n\), we write \(\varDelta _n \nearrow \varDelta \) (as \(n\rightarrow \infty \)). Denote by \(\text{ card}(X)\) the cardinal number of \(X\). If \(X\) is a topological space, then \(\mathfrak{B }(X)\) stands for the \(\sigma \)-algebra of Borel subsets of \(X\).

Let \(A\) be an operator in a complex Hilbert space \(\mathcal{H }\) (all operators considered in this paper are linear). Denote by \(\mathcal{D }(A), \mathcal N (A), \mathcal{R }(A), \bar{A}\) and \(A^*\) the domain, the kernel, the range, the closure and the adjoint of \(A\) (in case they exist). If \(A\) is closed and densely defined, then \(A\) has a (unique) polar decomposition \(A=U|A|\), where \(U\) is a partial isometry on \(\mathcal{H }\) such that \(\mathcal N (U)=\mathcal N (A)\) and \(|A|\) is the square root of \(A^*A\) (cf. [3, Section 8.1]). Set \(\mathcal{D }^\infty (A) = \bigcap _{n=0}^\infty \mathcal{D }(A^n)\). Members of \(\mathcal{D }^\infty (A)\) are called \(C^\infty \)-vectors of \(A\). Denote by \(\Vert \cdot \Vert _A\) the graph norm of \(A\), that is,

$$\begin{aligned} \Vert f\Vert _A^2 := \Vert f\Vert ^2 + \Vert Af\Vert ^2, \quad f \in \mathcal{D }(A). \end{aligned}$$

Given \(n \in \mathbb{Z }_+\), we define the norm \(\Vert \cdot \Vert _{A,n}\) on \(\mathcal{D }(A^n)\) by

$$\begin{aligned} \Vert f\Vert _{A,n}^2 := \sum _{j=0}^n \Vert A^j f\Vert ^2, \quad f \in \mathcal{D }(A^n). \end{aligned}$$

Clearly, for every \(n \in \mathbb N \), \((\mathcal{D }(A^n),\Vert \cdot \Vert _{A^n})\) and \((\mathcal{D }(A^n), \Vert \cdot \Vert _{A,n})\) are inner product spaces (with standard inner products). A vector subspace \(\mathcal{E }\) of \(\mathcal{D }(A)\) is called a core for \(A\) if \(\mathcal{E }\) is dense in \(\mathcal{D }(A)\) with respect to the graph norm of \(A\). Denote by \(I\) the identity operator on \(\mathcal{H }\).

By applying Propositions 2.1 and 3.2, one may obtain a criterion for closedness of a linear combination of composition operators.

Proposition 2.1

Let \(A_1, \ldots , A_n\) be closed operators in \(\mathcal{H }\) \((n\in \mathbb N )\). Then \(\sum _{j=1}^n A_j\) is closed if and only if there exists \(c\in \mathbb{R }_+\) such that

$$\begin{aligned} \sum _{j=1}^n \Vert A_j f\Vert ^2 {\leqslant }c \bigg (\Vert f\Vert ^2 + \left\Vert\sum _{j=1}^n A_j f\right\Vert^2\bigg ), \quad f \in \bigcap _{j=1}^n \mathcal{D }(A_j). \end{aligned}$$
(2.1)

Proof

Define the vector space \(\mathcal{X }=\bigcap _{j=1}^n \mathcal{D }(A_j)\) and the norm \(\Vert \cdot \Vert _{*}\) on \(\mathcal{X }\) by \(\Vert f\Vert _{*}^2 = \Vert f\Vert ^2 + \sum _{j=1}^n \Vert A_j f\Vert ^2\) for \(f \in \mathcal{X }\). Since the operators \(A_1, \ldots , A_n\) are closed, we deduce that \((\mathcal{X },\Vert \cdot \Vert _{*})\) is a Hilbert space. Recall that \(A:=\sum _{j=1}^n A_j\) is closed if and only if \((\mathcal{X },\Vert \cdot \Vert _{A})\) is a Hilbert space. Since the identity map from \((\mathcal{X },\Vert \cdot \Vert _{*})\) to \((\mathcal{X },\Vert \cdot \Vert _{A})\) is continuous, we conclude from the inverse mapping theorem that \((\mathcal{X },\Vert \cdot \Vert _{A})\) is a Hilbert space if and only if (2.1) holds for some \(c\in \mathbb{R }_+\). \(\square \)

A densely defined operator \(N\) in \(\mathcal{H }\) is said to be normal if \(N\) is closed and \(N^*N=NN^*\) (or equivalently if and only if \(\mathcal{D }(N)=\mathcal{D }(N^*)\) and \(\Vert Nf\Vert =\Vert N^*f\Vert \) for all \(f \in \mathcal{D }(N)\), see [55, Proposition, p. 125]). We say that a densely defined operator \(A\) in \(\mathcal{H }\) is formally normal if \(\mathcal{D }(A) \subseteq \mathcal{D }(A^*)\) and \(\Vert Af\Vert =\Vert A^*f\Vert \) for all \(f \in \mathcal{D }(A)\) (cf. [2, 12]). A densely defined operator \(A\) in \(\mathcal{H }\) is called hyponormal if \(\mathcal{D }(A) \subseteq \mathcal{D }(A^*)\) and \(\Vert A^*f\Vert {\leqslant }\Vert Af\Vert \) for all \(f \in \mathcal{D }(A)\) (cf. [27, 35, 53]). Clearly, a closed densely defined operator \(A\) in \(\mathcal{H }\) is normal if and only if both operators \(A\) and \(A^*\) are hyponormal. Obviously normality implies formal normality and formal normality implies hyponormality. It is well-known that none of these implications can be reversed in general. We say that a densely defined operator \(S\) in \(\mathcal{H }\) is subnormal if there exist a complex Hilbert space \(\mathcal{K }\) and a normal operator \(N\) in \(\mathcal{K }\) such that \(\mathcal{H }\subseteq \mathcal{K }\) (isometric embedding), \(\mathcal{D }(S) \subseteq \mathcal{D }(N)\) and \(Sf = Nf\) for all \(f \in \mathcal{D }(S)\).

The members of the next class are related to subnormal operators. A closed densely defined operator \(A\) in \(\mathcal{H }\) is said to be quasinormal if \(A\) commutes with the spectral measure \(E_{|A|}\) of \(|A|\), that is, \(E_{|A|}(\varDelta ) A \subseteq A E_{|A|}(\varDelta )\) for all \(\varDelta \in \mathfrak{B }(\mathbb{R }_+)\) (cf. [4, 49]). In view of [49, Proposition 1], a closed densely defined operator \(A\) in \(\mathcal{H }\) is quasinormal if and only if \(U |A| \subseteq |A|U\), where \(A=U|A|\) is the polar decomposition of \(A\). This combined with [3, Theorem 8.1.5] shows that if \(A\) is a normal operator, then \(A\) is quasinormal and \(\mathcal N (A)=\mathcal N (A^*)\). In turn, quasinormality together with the inclusion \(\mathcal N (A^*) \subseteq \mathcal N (A)\) characterizes normality. This result can be found in [52]. For the reader’s convenience, we include its proof.

Theorem 2.2

An operator \(A\) in \(\mathcal{H }\) is normal if and only if \(A\) is quasinormal and \(\mathcal N (A^*) \subseteq \mathcal N (A)\). Moreover, if \(A\) is normal, then \(\mathcal N (A) = \mathcal N (A^*)\).

Proof

In view of the above discussion, it is enough to prove the sufficiency. First, we show that if \(A\) is quasinormal and \(A = U|A|\) is its polar decomposition, then \(U |A| = |A|U\). Indeed, by [49, Proposition 1], \(U |A| \subseteq |A|U\). Taking adjoints, we get \(U^*|A|\subseteq |A|U^*\), which implies that \(U^*(\mathcal{D }(|A|)) \subseteq \mathcal{D }(|A|)\). Hence, if \(f \in \mathcal{D }(|A|U)\), then \(U^* Uf\in \mathcal{D }(|A|)\). Since \(I - U^* U\) is the orthogonal projection of \(\mathcal{H }\) onto \(\mathcal N (|A|)\), we conclude that \(f = U^* U f + (I - U^* U)f \in \mathcal{D }(|A|)\). This shows that \(\mathcal{D }(|A|U) \subset \mathcal{D }(U|A|)\), which implies that \(U |A| = |A|U\).

Now suppose that \(A\) is quasinormal and \(\mathcal N (A^*) \subseteq \mathcal N (A)\). Since the operators \(P := UU^*\) and \(P^\perp := (I-P)\) are the orthogonal projections of \(\mathcal{H }\) onto \(\overline{\mathcal{R }(A)}\) and \(\mathcal N (A^*)\), respectively, we infer from the inclusion \(\mathcal N (A^*) \subset \mathcal N (A)\) that

$$\begin{aligned} \mathcal{R }(P^\perp ) \subset \mathcal N (A)=\mathcal N (|A|) \subset \mathcal{D }(|A|^2). \end{aligned}$$
(2.2)

It follows from \(U|A|=|A|U\) and \(A^* = |A|U^*\) that

$$\begin{aligned} AA^* = U|A|^2U^* = |A|^2P. \end{aligned}$$
(2.3)

We will show that

$$\begin{aligned} |A|^2P = |A|^2. \end{aligned}$$
(2.4)

Indeed, if \(f\in \mathcal{H }\), then, by (2.2) and the equality \(f=Pf + P^\perp f\), we see that \(Pf\in \mathcal{D }(|A|^2)\) if and only if \(f \in \mathcal{D }(|A|^2)\). This implies that \(\mathcal{D }(|A|^2P) = \mathcal{D }(|A|^2)\). Using (2.2) again, we see that \(|A|^2 f = |A|^2 Pf\) for every \(f \in \mathcal{D }(|A|^2)\). Hence, the equality (2.4) is valid. Combining (2.3) with (2.4), we get \(AA^*=A^*A\).

The “moreover” part is well-known and easy to prove. \(\square \)

Recall that quasinormal operators are subnormal (see [4, Theorem 1] and [49, Theorem 2]). The reverse implication does not hold in general. Clearly, subnormal operators are hyponormal, but not reversely. It is worth pointing out that formally normal operators may not be subnormal (cf. [13, 40, 46]).

A finite complex matrix \([c_{i,j}]_{i,j=0}^n\) is said to be nonnegative if

$$\begin{aligned} \sum _{i,j=0}^n c_{i,j} \alpha _i \bar{\alpha }_j\, {\geqslant }\,\, 0, \quad \alpha _0,\ldots , \alpha _n \in \mathbb{C }. \end{aligned}$$

If this is the case, then we write \([c_{i,j}]_{i,j=0}^n\, {\geqslant }\,\, 0\). A sequence \(\{\gamma _n\}_{n=0}^\infty \) of real numbers is said to be a Stieltjes moment sequence if there exists a positive Borel measure \(\rho \) on \(\mathbb{R }_+\) such that

$$\begin{aligned} \gamma _n = \int \limits _{\mathbb{R }_+} s^n \mathrm{d}\rho (s), \quad n \in \mathbb{Z }_+. \end{aligned}$$

A sequence \(\{\gamma _n\}_{n=0}^\infty \subseteq \mathbb{R }\) is said to be positive definite if for every \(n\in \mathbb{Z }_+, [\gamma _{i+j}]_{i,j=0}^n {\geqslant }0\). By the Stieltjes theorem (see [1, Theorem 6.2.5]), we have

$$\begin{aligned}&\text{ a} \text{ sequence}\ \{{\gamma }_n\}_{n=0}^\infty \subseteq \mathbb{R }\ \text{ is} \text{ a} \text{ Stieltjes} \text{ moment} \text{ sequence} \text{ if} \text{ and} \text{ only} \text{ if} \nonumber \\&\text{ the} \text{ sequences}\ \{{\gamma }_n\}_{n=0}^\infty \ \text{ and}\ \{{\gamma }_{n+1}\}_{n=0}^\infty \ \text{ are} \text{ positive} \text{ definite}. \end{aligned}$$
(2.5)

Definition 2.3

We say that an operator \(S\) in \(\mathcal{H }\) generates Stieltjes moment sequences if \(\mathcal{D }^\infty (S)\) is dense in \(\mathcal{H }\) and \(\{\Vert S^n f\Vert ^2\}_{n=0}^\infty \) is a Stieltjes moment sequence for every \(f \in \mathcal{D }^\infty (S)\).

It is well-known that if \(S\) is subnormal, then \(\{\Vert S^n f\Vert ^2\}_{n=0}^\infty \) is a Stieltjes moment sequence for every \(f \in \mathcal{D }^\infty (S)\) (see [7, Proposition 3.2.1]; see also Proposition 2.4 below). Hence, if \(\mathcal{D }^\infty (S)\) is dense in \(\mathcal{H }\) and \(S\) is subnormal, then \(S\) generates Stieltjes moment sequences. It turns out that the converse implication does not hold in general (see [7, Section 3.2]).

The following can be proved analogously to [7, Proposition 3.2.1] by using (2.5).

Proposition 2.4

If \(S\) is a subnormal operator in \(\mathcal{H }\), then the following two assertions hold:

  1. (i)

    \(\big [\Vert S^{i+j}f\Vert ^2\big ]_{i,j=0}^n {\geqslant }0\) for all \(f\in \mathcal{D }(S^{2n})\) and \(n\in \mathbb{Z }_+\),

  2. (ii)

    \(\big [\Vert S^{i+j+1}f\Vert ^2\big ]_{i,j=0}^n {\geqslant }0\) for all \(f\in \mathcal{D }(S^{2n+1})\) and \(n\in \mathbb{Z }_+\).

For the reader’s convenience, we state a theorem which is occasionally called the Mittag-Leffler theorem (cf. [41, Lemma 1.1.2]).

Theorem 2.5

Let \(\left\{ \mathcal{E }_n \right\} _{n=0}^\infty \) be a sequence of Banach spaces such that for every \(n\in \mathbb{Z }_+, \mathcal{E }_{n+1}\) is a vector subspace of \(\mathcal{E }_n, \mathcal{E }_{n+1}\) is dense in \(\mathcal{E }_n\) and the embedding map of \(\mathcal{E }_{n+1}\) into \(\mathcal{E }_{n}\) is continuous. Then, \(\bigcap _{n=0}^\infty \mathcal{E }_n\) is dense in each space \(\mathcal{E }_k\)\(k\in \mathbb{Z }_+\).

3 Basic properties of composition operators

From now on, except for “Appendices A and B”, \((X, {\fancyscript{A}}, \mu )\) always stands for a \(\sigma \)-finite measure space. We shall abbreviate the expressions “almost everywhere with respect to \(\mu \)” and “for \(\mu \)-almost every \(x\)” to “a.e. \([\mu ]\)” and “for \(\mu \)-a.e. \(x\)”, respectively. As usual, \(L^2(\mu )=L^2(X,{\fancyscript{A}}, \mu )\) denotes the Hilbert space of all square integrable complex functions on \(X\). The norm and the inner product of \(L^2(\mu )\) are denoted by \(\Vert \cdot \Vert \) and \(\langle \cdot ,\text{-}\rangle \), respectively. Let \(\phi \) be an \({\fancyscript{A}}\)-measurable transformationFootnote 1 of \(X\), that is, \(\phi ^{-1}(\varDelta ) \in {\fancyscript{A}}\) for all \(\varDelta \in {\fancyscript{A}}\). Denote by \(\mu \circ \phi ^{-1}\) the positive measure on \({\fancyscript{A}}\) given by \(\mu \circ \phi ^{-1}(\varDelta )=\mu (\phi ^{-1}(\varDelta ))\) for all \(\varDelta \in {\fancyscript{A}}\). We say that \(\phi \) is nonsingular if \(\mu \circ \phi ^{-1}\) is absolutely continuous with respect to \(\mu \). It is easily seen that if \(\phi \) is nonsingular, then the mapping \(C_\phi :L^2(\mu ) \supseteq \mathcal{D }(C_\phi )\rightarrow L^2(\mu )\) given by

$$\begin{aligned} \mathcal{D }(C_\phi ) = \{f \in L^2(\mu ) :f \circ \phi \in L^2(\mu )\} \text{ and} C_\phi f = f \circ \phi \text{ for} f \in \mathcal{D }(C_\phi ), \end{aligned}$$
(3.1)

is well-defined and linear. Such an operator is called a composition operator induced by \(\phi \); the transformation \(\phi \) will be referred to as a symbol of \(C_\phi \). Note that if the operator \(C_\phi \) given by (3.1) is well-defined, then the transformation \(\phi \) is nonsingular.

Convention. For the remainder of this paper, whenever \(C_{\phi }\) is mentioned the transformation \(\phi \) is assumed to be nonsingular.

If \(\phi \) is nonsingular, then by the Radon-Nikodym theorem there exists a unique (up to sets of measure zero) \({\fancyscript{A}}\)-measurable function \(\mathsf{h }_\phi :X \rightarrow [0,\infty ]\) such that

$$\begin{aligned} \mu \circ \phi ^{-1}(\varDelta ) = \int \limits _\varDelta \mathsf{h }_\phi d\mu , \quad \varDelta \in {\fancyscript{A}}. \end{aligned}$$
(3.2)

Here and later on \(\phi ^n\) stands for the \(n\)-fold composition of \(\phi \) with itself if \(n {\geqslant }1\) and \(\phi ^0\) for the identity transformation of \(X\). We also write \(\phi ^{-n} (\varDelta ) := (\phi ^n)^{-1}(\varDelta )\) for \(\varDelta \in {\fancyscript{A}}\) and \(n\in \mathbb{Z }_+\). Note that \(\mathsf{h }_{\phi ^0} =1\) a.e. \([\mu ]\). It is clear that the composition \(\phi _1 \circ \ldots \circ \phi _n\) of finitely many nonsingular transformations \(\phi _1, \ldots , \phi _n\) of \(X\) is a nonsingular transformation and

$$\begin{aligned} C_{\phi _n} \ldots C_{\phi _1} \subseteq C_{\phi _1\circ \ldots \circ \phi _n}, \quad n \in \mathbb N . \end{aligned}$$
(3.3)

Now we construct an \({\fancyscript{A}}\)-measurable transformation \(\phi \) of \(X\) such that \(\phi \) is not nonsingular while \(\phi ^2\) is nonsingular.

Example 3.1

Set \(X=\{0\}\cup \{1\}\cup [2,3]\). Let \({\fancyscript{A}}=\{\varDelta \cap X:\varDelta \in \mathfrak{B }(\mathbb{R }_+)\}\). Define the finite Borel measure \(\mu \) on \(X\) by

$$\begin{aligned} \mu (\varDelta )= \chi _\varDelta (0) + \chi _\varDelta (1) + m(\varDelta \cap [2,3]), \quad \varDelta \in {\fancyscript{A}}, \end{aligned}$$

where \(m\) stands for the Lebesgue measure on \(\mathbb{R }\). Let \(\phi \) be an \({\fancyscript{A}}\)-measurable transformation of \(X\) given by \(\phi (0)=2, \phi (1)=1\) and \(\phi (x)=1\) for \(x \in [2,3]\). Since \(\mu (\{2\})=0\) and \((\mu \circ \phi ^{-1})(\{2\})=1\), we see that \(\phi \) is not nonsingular. However, \(\phi ^2\) is nonsingular because \(\phi ^2(x)=1\) for all \(x\in X\) and \(\mu (\{1\})>0\).

Suppose that \(\phi \) is a nonsingular transformation of \(X\). In view of the measure transport theorem ([22, Theorem C, p. 163]), we have

$$\begin{aligned} \int \limits _X |f \circ \phi |^2 \mathrm{d}\mu = \int \limits _X |f|^2 \mathsf{h }_\phi \mathrm{d}\mu \text{ for} \text{ every} {\fancyscript{A}}\text{-measurable} \text{ function} f:X \rightarrow \mathbb{C }. \end{aligned}$$
(3.4)

This implies that

$$\begin{aligned} \mathcal{D }(C_\phi ) = L^2((1 + \mathsf{h }_\phi )\mathrm{d}\mu ), \quad \Vert f\Vert _{C_\phi }^2 = \int \limits _X |f|^2 (1 + \mathsf{h }_\phi ) \mathrm{d}\mu ,\end{aligned}$$
(3.5)
$$\begin{aligned} \mathcal{D }(C_{\phi }^n) = L^2\Big (\Big (\sum _{j=0}^n \mathsf{h }_{\phi ^j}\Big )\mathrm{d}\mu \Big ), \quad \Vert f\Vert _{C_\phi ,n}^2 = \int \limits _X |f|^2 \Big (\sum _{j=0}^n \mathsf{h }_{\phi ^j}\Big ) \mathrm{d}\mu , \quad n \in \mathbb{Z }_+. \end{aligned}$$
(3.6)

Moreover, if \(\phi _1, \ldots , \phi _n\) are nonsingular transformations of \(X\) (\(n \in \mathbb N \)), then

$$\begin{aligned} \mathcal{D }(C_{\phi _n} \ldots C_{\phi _1}) = L^2((1 + \sum _{j=1}^n \mathsf{h }_{\phi _1\circ \cdots \circ \phi _j})\mathrm{d}\mu ). \end{aligned}$$
(3.7)

The following proposition is somewhat related to [17, p. 664] and [11, Lemma 6.1].

Proposition 3.2

Let \(\phi \) be a nonsingular transformation of \(X\). Then \(C_\phi \) is a closed operator and

$$\begin{aligned} \overline{\mathcal{D }(C_\phi )} = \chi _{\mathsf{F }_{\phi }} L^2(\mu ) \text{ with} \mathsf F _{\phi } = \big \{x \in X:\mathsf{h }_\phi (x) < \infty \big \}\text{.} \end{aligned}$$
(3.8)

Moreover, the following conditions are equivalent:

  1. (i)

    \(C_\phi \) is densely defined,

  2. (ii)

    \(\mathsf{h }_\phi < \infty \) a.e. \([\mu ]\),

  3. (iii)

    the measure \(\mu \circ \phi ^{-1}\) is \(\sigma \)-finite.

Proof

Applying (3.5), we get \(C_\phi = \overline{C_\phi }\) and \(\overline{\mathcal{D }(C_\phi )} \subseteq \chi _{\mathsf{F }_{\phi }} L^2(\mu )\). To prove the opposite inclusion \(\chi _{\mathsf{F }_{\phi }} L^2(\mu )\subseteq \overline{\mathcal{D }(C_\phi )}\), take \(f\in L^2(\mu )\) such that \(f|_{X \setminus \mathsf{F }_\phi } = 0\) a.e. \([\mu ]\), and set \(X_{n}=\{x \in X:\mathsf{h }_\phi (x) {\leqslant }n\}\) for \(n \in \mathbb N \). Noting that \(X_n \nearrow \mathsf{F }_\phi \) as \(n\rightarrow \infty \), we see that \(\int _{X} |\chi _{X_{n}}f|^2 (1+\mathsf{h }_\phi ) \mathrm{d}\mu < \infty \) for all \(n\in \mathbb N \), and \(\lim _{n\rightarrow \infty }\int _X|f - \chi _{X_n}f|^2 \mathrm{d}\mu =0\), which completes the proof of (3.8).

  • (i)\(\Leftrightarrow \)(ii) Employ (3.8).

  • (ii)\(\Leftrightarrow \)(iii) Apply (3.2) and the assumption that \(\mu \) is \(\sigma \)-finite.\(\square \)

Corollary 3.3

Suppose that \(\phi _1, \ldots , \phi _n\) are nonsingular transformations of \(X\) and \(\lambda _1, \ldots , \lambda _n\) are nonzero complex numbers \((n\in \mathbb N )\). Then, \(\sum _{j=1}^n \lambda _j C_{\phi _j}\) is densely defined if and only if \(C_{\phi _k}\) is densely defined for every \(k=1, \ldots , n\).

Proof

By (3.5), \(\mathcal{D }(\sum _{j=1}^n \lambda _j C_{\phi _j}) = L^2((1 + \sum _{j=1}^n \mathsf{h }_{\phi _j})\mathrm{d}\mu )\), and thus the “if” part follows from Proposition 3.2 and Lemma 12.1. The “only if” part is obvious. \(\square \)

4 Products of composition operators

First we give necessary and sufficient conditions for a product of composition operators to be densely defined.

Proposition 4.1

Let \(\phi _1, \ldots , \phi _n\) be nonsingular transformations of \(X\) \((2 {\leqslant }n < \infty )\). Then, the following assertions hold:

  1. (i)

    \(C_{\phi _n} \ldots C_{\phi _1}\) is a closable operator,

  2. (ii)

    \(C_{\phi _n} \ldots C_{\phi _1}\) is densely defined if and only if \(C_{\phi _1\circ \ldots \circ \phi _k}\) is densely defined for every \(k=1, \ldots , n\),

  3. (iii)

    if \(C_{\phi _{n-1}} \ldots C_{\phi _1}\) is densely defined, then

    $$\begin{aligned} C_{\phi _1\circ \ldots \circ \phi _k} = \overline{C_{\phi _k} \ldots C_{\phi _1}}, \quad k=1, \ldots , n, \end{aligned}$$
    (4.1)
  4. (iv)

    if \(C_{\phi _1 \circ \ldots \circ \phi _n}\) is densely defined, then so is the operator \(C_{\phi _n}\),

  5. (v)

    if \(C_{\phi _n} \ldots C_{\phi _1}\) is densely defined, then so are the operators \(C_{\phi _1}\), ..., \(C_{\phi _n}\).

Proof

  1. (i)

    Apply (3.3) and Proposition 3.2.

  2. (ii)

    To prove the “if” part, assume that \(C_{\phi _1\circ \ldots \circ \phi _k}\) is densely defined for \(k=1, \ldots , n\). It follows from Proposition 3.2 that \(\mathsf{h }_{\phi _1\circ \ldots \circ \phi _k} < \infty \) a.e. \([\mu ]\) for \(k=1,\ldots , n\). Applying (3.7) and Lemma 12.1 to \(\rho _1\equiv 1\) and \(\rho _2 = 1 + \sum _{j=1}^n \mathsf{h }_{\phi _1\circ \ldots \circ \phi _j}\) we get \(\overline{\mathcal{D }(C_{\phi _n} \ldots C_{\phi _1})}=L^2(\mu )\). The “only if” part follows from (3.3) and the fact that the operators \(C_{\phi _k} \ldots C_{\phi _1}, k=1, \ldots , n\), are densely defined.

  3. (iii)

    It follows from (ii) and Proposition 3.2 that \(h:=\sum _{j=1}^{n-1} \mathsf{h }_{\phi _1\circ \ldots \circ \phi _j} < \infty \) a.e. \([\mu ]\). Set \(Y=\{x\in X :\mathsf{h }_{\phi _1\circ \ldots \circ \phi _n} (x) < \infty \}\) and \({\fancyscript{A}}_Y = \{\varDelta \in {\fancyscript{A}}:\varDelta \subseteq Y\}\). Equip \(\mathcal{D }(C_{\phi _1\circ \ldots \circ \phi _n})\) with the graph norm of \(C_{\phi _1\circ \ldots \circ \phi _n}\) and note that the mapping

    $$\begin{aligned} \varTheta :\mathcal{D }(C_{\phi _1\circ \ldots \circ \phi _n}) \ni f \longmapsto f|_{Y} \in L^2\big (Y,{\fancyscript{A}}_Y,(1+\mathsf{h }_{\phi _1\circ \ldots \circ \phi _n})\mathrm{d}\mu \big ) \end{aligned}$$

    is a well-defined unitary isomorphism (use (3.5)). It follows from Lemma 12.1 that \(L^2\big (Y,{\fancyscript{A}}_Y,(1+ h + \mathsf{h }_{\phi _1\circ \ldots \circ \phi _n})\mathrm{d}\mu \big )\) is dense in \(L^2\big (Y,{\fancyscript{A}}_Y,(1+\mathsf{h }_{\phi _1\circ \ldots \circ \phi _n})\mathrm{d}\mu \big )\). Since, by (3.3) and (3.7), \(\varTheta (\mathcal{D }(C_{\phi _{n}} \ldots C_{\phi _{1}}))=L^2\big (Y,{\fancyscript{A}}_Y,(1+ h + \mathsf{h }_{\phi _1\circ \ldots \circ \phi _n})\mathrm{d}\mu \big )\), we deduce that \(\overline{C_{\phi _n} \ldots C_{\phi _1}} = C_{\phi _1\circ \ldots \circ \phi _n}\). Applying the previous argument to the systems \((C_{\phi _1}, \ldots , C_{\phi _k}), k\in \{1, \ldots , n-1\}\), we obtain (4.1).

  4. (iv)

    It is sufficient to discuss the case of \(n=2\). Suppose that \(C_{\phi _1\circ \phi _2}\) is densely defined. In view of Proposition 3.2, the measure \(\mu \circ (\phi _1 \circ \phi _2)^{-1}\) is \(\sigma \)-finite. Since \(\mu \circ (\phi _1 \circ \phi _2)^{-1} = (\mu \circ \phi _2^{-1})\circ \phi _1^{-1}\), we see that the measure \(\mu \circ \phi _2^{-1}\) is \(\sigma \)-finite as well. Applying Proposition 3.2 again, we conclude that \(C_{\phi _2}\) is densely defined.

  5. (v)

    Apply (ii) and (iv). \(\square \)

Corollary 4.2

If \(C_{\phi }^{n-1}\) is densely defined for some \(n\in \mathbb N \), then \(\overline{C_{\phi }^n} = C_{\phi ^n}\).

The following is an immediate consequence of (3.7) and Corollary 12.4.

Proposition 4.3

If \(\phi _1, \ldots , \phi _m\) and \(\psi _1, \ldots , \psi _n\) are nonsingular transformations of \(X\), then \(\mathcal{D }(C_{\phi _n} \ldots C_{\phi _1}) \subseteq \mathcal{D }(C_{\psi _m} \ldots C_{\psi _1})\) if and only if there exists \(c \in \mathbb{R }_+\) such that \(\sum _{j=1}^m \mathsf{h }_{\psi _1\circ \ldots \circ \psi _j} {\leqslant }c \big (1+ \sum _{j=1}^n \mathsf{h }_{\phi _1\circ \ldots \circ \phi _j}\big )\) a.e. \([\mu ]\).

Now we give necessary and sufficient conditions for a product of composition operators to be closed.

Proposition 4.4

Let \(\phi _1, \ldots , \phi _n\) be nonsingular transformations of \(X\) \((2 \le n < \infty )\). Then, the following three conditions are equivalent:

  1. (i)

    \(C_{\phi _n} \ldots C_{\phi _1} = C_{\phi _1\circ \ldots \circ \phi _n}\),

  2. (ii)

    \(\mathcal{D }(C_{\phi _1\circ \ldots \circ \phi _n}) \subseteq \mathcal{D }(C_{\phi _n} \ldots C_{\phi _1})\),

  3. (iii)

    there exists \(c\in \mathbb{R }_+\) such that \(\sum _{j=1}^{n-1} \mathsf{h }_{\phi _1\circ \ldots \circ \phi _j} {\leqslant }c (1+\mathsf{h }_{\phi _1\circ \ldots \circ \phi _n})\) a.e. \([\mu ]\).

Moreover, any of the conditions (i) to (iii) implies that

  1. (iv)

    \(C_{\phi _n} \ldots C_{\phi _1}\) is closed.

If \(C_{\phi _{n-1}} \ldots C_{\phi _1}\) is densely defined, then all the conditions (i) to (iv) are equivalent.

Proof

The equivalence of (i) and (ii) is a direct consequence of (3.3). The equivalence of (ii) and (iii) follows from Proposition 4.3. That (i) implies (iv) follows from Proposition 3.2. Finally, if the product \(C_{\phi _{n-1}} \ldots C_{\phi _1}\) is densely defined, then (iv) implies (i) due to Proposition 4.1 (iii). \(\square \)

Corollary 4.5

If \(\phi \) is a nonsingular transformation of \(X\), then the following assertions hold for all \(n \in \mathbb N \) :

  1. (i)

    \(C_{\phi ^n}\) is densely defined if and only if \(\mathsf{h }_{\phi ^n} < \infty \) a.e. \([\mu ]\),

  2. (ii)

    \(C_{\phi }^n\) is densely defined if and only if \(\sum _{j=1}^n \mathsf{h }_{\phi ^j} < \infty \) a.e. \([\mu ]\),

  3. (iii)

    \(C_{\phi }^n=C_{\phi ^n}\) if and only if there exists \(c\in \mathbb{R }_+\) such that \(\mathsf{h }_{\phi ^k} {\leqslant }c (1 + \mathsf{h }_{\phi ^n})\) a.e. \([\mu ]\) for \(k=1, \ldots , n\).

Proof

Use Propositions 3.2, 4.1 (ii) and 4.4 (for (ii) see also [24, p. 515]). \(\square \)

Corollary 4.6

If \(\phi \) is a nonsingular transformation of \(X\) and \(\overline{\mathcal{D }(C_\phi ^m)}=L^2(\mu )\) for some \(m \in \mathbb N \), then there exists a sequence \(\{X_n\}_{n=1}^\infty \subseteq {\fancyscript{A}}\) such that

  1. (i)

    \(X_n\nearrow X\) as \(n\rightarrow \infty \),

  2. (ii)

    \(\mu (X_n) < \infty \) for all \(n\in \mathbb N \),

  3. (iii)

    \(\sum _{j=1}^m \mathsf{h }_{\phi ^j}(x) {\leqslant }n\) for \(\mu \)-a.e. \(x \in X_n\) and \(n\in \mathbb N \).

The question of when \(C^\infty \)-vectors of an operator \(A\) in a Hilbert space \(\mathcal{H }\) form a dense subspace of \(\mathcal{H }\) is of independent interest (cf. [28, 39]). If every power of \(A\) is densely defined, then one could expect that \(\mathcal{D }^\infty (A)\) is dense in \(\mathcal{H }\). This is the case for any closed densely defined operator (even in a Banach space), the resolvent set of which is nonemptyFootnote 2. As shown below, this is also the case for composition operators. However, this seems to be not true in general. Dropping the assumption of closedness, we can provide a simple counterexample. Indeed, take an infinite dimensional separable Hilbert space \(\mathcal{H }\). Then, there exists a dense subset \(\{e_n:n \in \mathbb{Z }_+\}\) of \(\mathcal{H }\) which consists of linearly independent vectors. Let \(A\) be the operator in \(\mathcal{H }\) whose domain is the linear span of \(\{e_n:n \in \mathbb N \}\) and \(Ae_j = e_{j-1}\) for every \(j\in \mathbb N \). Since \(\{e_n:n {\geqslant }k\}\) is dense in \(\mathcal{H }\) for every \(k \in \mathbb{Z }_+\), we deduce that the operator \(A^n\) is densely defined for every \(n \in \mathbb{Z }_+\). However, \(\mathcal{D }^\infty (A)=\{0\}\).

Theorem 4.7

If \(\phi \) is a nonsingular transformation of \(X\), then the following conditions are equivalent:

  1. (i)

    \(\mathcal{D }(C_\phi ^n)\) is dense in \(L^2(\mu )\) for every \(n \in \mathbb N \),

  2. (ii)

    \(\mathcal{D }^\infty (C_\phi )\) is dense in \(L^2(\mu )\),

  3. (iii)

    \(\mathcal{D }^\infty (C_\phi )\) is a core for \(C_\phi ^n\) for every \(n\in \mathbb{Z }_+\),

  4. (iv)

    \(\mathcal{D }^\infty (C_\phi )\) is dense in \((\mathcal{D }(C_\phi ^n),\Vert \cdot \Vert _{C_\phi ,n})\) for every \(n\in \mathbb{Z }_+\).

Proof

The implications (iv)\(\Rightarrow \)(iii), (iii)\(\Rightarrow \)(ii) and (ii)\(\Rightarrow \)(i) are obvious.

(i)\(\Rightarrow \)(iv) In view of Corollary 4.5 (ii), \(0 {\leqslant }\mathsf{h }_{\phi ^n} < \infty \) a.e. \([\mu ]\) for all \(n\in \mathbb N \). Given \(n \in \mathbb{Z }_+\) we denote by \(\mathcal{H }_n\), the inner product space \((\mathcal{D }(C_\phi ^n), \Vert \cdot \Vert _{C_\phi ,n})\). It follows from (3.6) that \(\mathcal{H }_n\) is a Hilbert space which coincides with \(L^2((\sum _{j=0}^n \mathsf{h }_{\phi ^j})\mathrm{d}\mu )\). Hence, in view of Lemma 12.1, \(\mathcal{H }_{n+1}\) is a dense subspace of \(\mathcal{H }_{n}\). Clearly, the embedding map of \(\mathcal{H }_{n+1}\) into \(\mathcal{H }_n\) is continuous. Applying Theorem 2.5 to the sequence \(\left\{ \mathcal{H }_n \right\} _{n=0}^\infty \), we conclude that \(\mathcal{D }^\infty (C_\phi ) = \bigcap _{i=0}^\infty \mathcal{H }_i\) is dense in \(\mathcal{D }(C_\phi ^n)\) with respect to the norm \(\Vert \cdot \Vert _{C_\phi ,n}\) for every \(n\in \mathbb{Z }_+\). This completes the proof. \(\square \)

Regarding Theorem 4.7, we mention the following surprising fact which can be deduced from [39, Theorem 4.5] by using Theorem 2.5 and [39, Corollaries 1.2 and 1.4].

Theorem 4.8

Let \(A\) be an unbounded selfadjoint operator in a complex Hilbert space \(\mathcal{H }\) and let \(\mathfrak N \) be a \((\)possibly empty\()\) subset of \(\mathbb N \setminus \{1\}\) such that \(\mathbb N \setminus \mathfrak N \) is infinite. Then, there exists a closed symmetric operator \(T\) in \(\mathcal{H }\) such that \(T \subseteq A, \mathcal{D }^\infty (T)\) is dense in \(\mathcal{H }\) and for every \(k\in \mathbb N , \mathcal{D }^\infty (T)\) is a core for \(T^k\) if and only if \(k \in \mathbb N \setminus \mathfrak N \).

5 Examples

We begin by showing that Corollary 4.2 is no longer true if the assumption that \(C_{\phi }^{n-1}\) is densely defined is dropped.

Example 5.1

We will demonstrate that there is a nonsingular transformation \(\phi \) such that \(C_{\phi }\) is densely defined, \(C_{\phi ^j}\) and \(C_{\phi }^j\) are not densely defined for every \(j \in \{2,3, \ldots \}\), and \(\overline{C_{\phi }^3} \not \subseteq C_{\phi ^3}\) (however, by Corollary 4.2, \(\overline{C_{\phi }^2} = C_{\phi ^2}\)). For this, we will re-examine Example 4.2 given in [24]. Suppose that \(\{a_i\}_{i=0}^\infty , \{b_i\}_{i=0}^\infty \) and \(\{c_{i,j}\}_{i,j=0}^\infty \) are disjoint sets of distinct elements. Set \(X=\{a_i\}_{i=0}^\infty \cup \{b_i\}_{i=0}^\infty \cup \{c_{i,j}\}_{i,j=0}^\infty \) and \({\fancyscript{A}}=2^X\). Let \(\mu \) be a unique \(\sigma \)-finite measure on \({\fancyscript{A}}\) determined by

$$\begin{aligned} \mu \big (\{x\}\big ) = {\left\{ \begin{array}{ll} 1&\text{ if} x=a_i \quad \text{ for} \text{ some} i\in \mathbb{Z }_+,\\ \frac{1}{2^{i+1}}&\quad \text{ if} x=b_i \text{ for} \text{ some} i\in \mathbb{Z }_+,\\ \frac{1}{2^{j+1}}&\quad \text{ if} x=c_{i,j} \text{ for} \text{ some} i,j\in \mathbb{Z }_+. \end{array}\right.} \end{aligned}$$

Define a nonsingular transformation \(\phi \) of \(X\) by

$$\begin{aligned} \phi (x)= {\left\{ \begin{array}{ll} a_{i+1}&\text{ if} x=a_i \quad \text{ for} \text{ some} i\in \mathbb{Z }_+,\\ a_0&\text{ if} x=b_i \quad \text{ for} \text{ some} i\in \mathbb{Z }_+,\\ b_i&\text{ if} x=c_{i,j}\quad \text{ for} \text{ some} i,j\in \mathbb{Z }_+. \end{array}\right.} \end{aligned}$$

Then \(\mathsf{h }_\phi < \infty \) a.e. \([\mu ]\), and thus by Proposition 3.2, the operator \(C_\phi \) is densely defined. Since \(\mathsf{h }_{\phi ^2}(a_0)=\infty \), we infer from Proposition 3.2 that \(C_{\phi ^2}\) is not densely defined. It follows from (3.7) that \(\mathcal{D }(C_{\phi }^3) = L^2((1 + \mathsf{h }_\phi + \mathsf{h }_{\phi ^2} + \mathsf{h }_{\phi ^3})\mathrm{d}\mu )\). This and \(\mathsf{h }_{\phi ^2}(a_0)=\infty \) imply that \(f(a_0)=0\) for every \(f \in \mathcal{D }(C_{\phi }^3)\). Since the convergence in the graph norm is stronger than the pointwise convergence, we deduce that \(f(a_0)=0\) for every \(f \in \mathcal{D }(\overline{C_{\phi }^3})\). As \(\mathcal{D }(C_{\phi ^3}) = L^2((1 + \mathsf{h }_{\phi ^3})\mathrm{d}\mu )\) (cf. (3.5)) and \(\mathsf{h }_{\phi ^3}(a_0)=0\) (because \(\phi ^{-3}(\{a_0\})=\varnothing \)), we see that \(\chi _{\{a_0\}} \in \mathcal{D }(C_{\phi ^3}) \setminus \mathcal{D }(\overline{C_{\phi }^3})\). Finally, arguing as above and using the fact that \(\mathsf{h }_{\phi ^{j+2}}(a_j)=\infty \) for every \(j\in \mathbb{Z }_+\), we conclude that \(C_{\phi ^j}\) is not densely defined for every \(j\in \{2,3, \ldots \}\). As a consequence, \(C_{\phi }^j\) is not densely defined for every \(j \in \{2,3, \ldots \}\).

The composition operator \(C_\phi \) constructed in Example 5.1 is densely defined, and its square is not densely defined; however, \(\dim \mathcal{D }(C_\phi ^n)= \infty \) for all \(n\in \mathbb N \) (because \(\chi _{\{a_{i}\}} \in \mathcal{D }(C_\phi ^n)\) for all \(i{\geqslant }n-1\)). In fact, there are more pathological examples.

Example 5.2

It was proved in [26, Theorem 4.2] that there exists a hyponormal weighted shift \(S\) on a rootless and leafless directed tree with positive weights whose square has trivial domain. By [25, Lemma 4.3.1], \(S\) is unitarily equivalent to a composition operator \(C\). As a consequence, \(C\) is injective and hyponormal, and \(\mathcal{D }(C^2) = \mathcal{D }^\infty (C)=\{0\}\) (see also [6] for a recent construction).

Regarding Proposition 4.1, we note that it may happen that the operators \(C_{\phi _1}\) and \(C_{\phi _2}\) are densely defined, while the operators \(C_{\phi _1\circ \phi _2}\) and \(C_{\phi _2}C_{\phi _1}\) are not (even if \(\phi _1=\phi _2\), see Example 5.1). Below, we will show that for some \(\phi _1\) and \(\phi _2\) the composition operator \(C_{\phi _1\circ \phi _2}\) is densely defined (even bounded), while \(C_{\phi _1}\) is not.

Example 5.3

Set \(X=\mathbb{Z }_+\) and \({\fancyscript{A}}=2^X\). Let \(\mu \) be the counting measure on \(X\) and let \(\phi _1\) and \(\phi _2\) be the nonsingular transformations of \(X\) given by \(\phi _1(2n)=n, \phi _1(2n+1)=0\) and \(\phi _2(n)=2n\) for \(n\in \mathbb{Z }_+\). Then, \(\phi _1\circ \phi _2\) is the identity transformation of \(X\), and hence, \(C_{\phi _1\circ \phi _2}\) is the identity operator on \(L^2(\mu )\). However, since \(\mu (\phi _1^{-1}(\{0\})) = \infty \), the measure \(\mu \circ \phi _1^{-1}\) is not \(\sigma \)-finite, and thus by Proposition 3.2, the operator \(C_{\phi _1}\) is not densely defined.

Our next aim is to provide examples showing that the equality \(C_{\phi }^n=C_{\phi ^n}\) which appears in Corollary 4.5 (iii) does not hold in general even if \(\mathcal{D }^\infty (C_\phi )\) is dense in \(L^2(\mu )\) (which is not the case for the operator given in Example 5.1).

Example 5.4

Set \(X=\mathbb N \) and \({\fancyscript{A}}=2^X\). Let \(\mu \) be a counting measure on \(X\) and let \(\{J_n\}_{n=1}^\infty \) be a partition of \(X\). Define a nonsingular transformation \(\phi \) of \(X\) by \(\phi (x) = \min J_{n^2}\) for \(x \in J_n\) and \(n\in \mathbb N \). Set \(\mathbb N _\mathrm{s}= \{n^2:n \in \mathbb N \}\) and note that

$$\begin{aligned} X=\{1\} \sqcup \bigsqcup _{q\in \mathbb N \setminus \mathbb N _\mathrm{s}} \big \{q^{2^n}:n \in \mathbb{Z }_+\big \}, \end{aligned}$$
(5.1)

where all terms in (5.1) are pairwise disjoint (they are equivalence classes under the equivalence relation \(\sim \) given by: \(p \sim q\) if and only if \(p^{2^m} = q^{2^n}\) for some \(m,n\in \mathbb{Z }_+\)). Since \(\mathsf{h }_{\phi ^j}(x) = \text{ card}(\phi ^{-j}(\{x\}))\) for \(x \in X\) and \(j\in \mathbb N \), we infer from (5.1) that for all \(j\in \mathbb N \) and \(x\in X\) (\(m\) appearing below varies over the set of integers)

$$\begin{aligned} \mathsf{h }_{\phi ^j}(x) = {\left\{ \begin{array}{ll} \text{ card}(J_1)&\text{ if} x=\min J_1,\\ \text{ card}(J_{q^{2^{m-j}}})&\text{ if} x=\min J_{q^{2^{m}}} \text{ with} q \in \mathbb N \setminus \mathbb N _\mathrm{s} \text{ and} m{\geqslant }j,\\ 0&\text{ otherwise}. \end{array}\right.} \end{aligned}$$
(5.2)

By (5.1), (5.2), Proposition 3.2 and Theorem 4.7, the following are equivalent:

  • \(\text{ card}(J_k) < \aleph _0\) for every \(k\in \mathbb N \),

  • \(C_\phi \) is densely defined,

  • \(C_\phi ^n\) is densely defined for some \(n\in \mathbb N \),

  • \(C_\phi ^n\) is densely defined for every \(n\in \mathbb N \),

  • \(\mathcal{D }^\infty (C_\phi )\) is dense in \(L^2(\mu )\).

The above, combined with (5.2) and Proposition 4.4, implies that if \(C_{\phi }\) is densely defined, then for every integer \(n{\geqslant }2, C_{\phi }^n\) is closed if and only if there exists \(c\in \mathbb{R }_+\) such that

$$\begin{aligned} \text{ card}(J_{q^{2^s}})&{\leqslant }&c, \quad s=0, \ldots , n-2,\, q \in \mathbb N \!\setminus \!\mathbb N _\mathrm{s},\\ \text{ card}(J_{q^{2^{s+1}}})&{\leqslant }&c \big (1+ \text{ card}(J_{q^{2^{s}}})\big ), \quad s\in \mathbb{Z }_+,\, q \in \mathbb N \!\setminus \! \mathbb N _\mathrm{s}. \end{aligned}$$

Using this and an induction argument, one can prove that if \(C_{\phi }\) is densely defined, then either \(C_\phi ^n\) is closed for every integer \(n{\geqslant }1\), or \(C_\phi ^n\) is not closed for every integer \(n{\geqslant }2\). Summarizing, if we choose a partition \(\{J_i\}_{i=1}^\infty \) of \(X\) such that \(J_n\) is finite for every \(n\in \mathbb N \), and \(\sup \{ \text{ card}(J_q) :q\in \mathbb N \!\setminus \! \mathbb N _\mathrm{s}\} = \aleph _0\) (which is possible), then \(\mathcal{D }^\infty (C_\phi )\) is dense in \(L^2(\mu )\) and \(C_\phi ^n\) is not closed for every integer \(n{\geqslant }2\). On the other hand, if \(\kappa {\geqslant }2\) is any fixed integer and a partition \(\{J_i\}_{i=1}^\infty \) of \(X\) is selected so that \(J_1\) is finite and \(\text{ card}(J_{q^{2^{n}}})=\kappa ^n\) for all \(n\in \mathbb{Z }_+\) and \(q \in \mathbb N \setminus \mathbb N _\mathrm{s}\) (which is also possible), then \(\mathcal{D }^\infty (C_\phi )\) is dense in \(L^2(\mu )\) and \(C_\phi ^n\) is closed and unbounded for every \(n\in \mathbb N \).

6 Injectivity of \(C_\phi \)

In this section, we provide necessary and sufficient conditions for a composition operator to be injective. The following set plays an important role in our considerations.

$$\begin{aligned} \mathsf N _\phi =\{x \in X :\mathsf{h }_\phi (x)=0\}. \end{aligned}$$

The following description of the kernel of \(C_\phi \) follows immediately from (3.4).

Proposition 6.1

If \(\phi :X \rightarrow X\) is nonsingular, then \(\mathcal N (C_\phi ) = \chi _{\mathsf{N }_\phi } L^2(\mu )\).

Proposition 6.2

Let \(\phi \) be a nonsingular transformation of \(X\). Consider the following four conditions:

  1. (i)

    \(\mathcal N (C_\phi ) = \{0\}\),

  2. (ii)

    \(\mu (\mathsf{N }_\phi )=0\),

  3. (iii)

    \(\chi _{\mathsf{N }_\phi } \circ \phi = \chi _{\mathsf{N }_\phi }\) a.e. \([\mu ]\),

  4. (iv)

    \(\mathcal N (C_\phi ) \subseteq \mathcal N (C_\phi ^*)\).

Then the conditions (i), (ii) and (iii) are equivalent. Moreover, if \(C_\phi \) is densely defined, then the conditions (i) to (iv) are equivalent.

Proof

(i)\(\Leftrightarrow \)(ii) Apply Proposition 6.1 and the \(\sigma \)-finiteness of \(\mu \).

(ii)\(\Rightarrow \)(iii) Since \(\phi \) is nonsingular, we have \(\mu (\mathsf{N }_\phi )=0\) and \(\mu (\phi ^{-1}(\mathsf{N }_\phi ))=0\), which implies that \(\mu (\mathsf{N }_\phi \vartriangle \phi ^{-1}(\mathsf{N }_\phi ))=0\). The latter is equivalent to (iii).

(iii)\(\Rightarrow \)(ii) By the measure transport theorem, we have

$$\begin{aligned} \mu (\mathsf{N }_\phi ) = \int \limits _X \chi _{\mathsf{N }_\phi } \mathrm{d}\mu = \int \limits _X \chi _{\mathsf{N }_\phi } \circ \phi \mathrm{d}\mu = \int \limits _X \chi _{\mathsf{N }_\phi } \mathsf{h }_\phi \mathrm{d}\mu =0. \end{aligned}$$

Now suppose that \(C_\phi \) is densely defined.

(i)\(\Rightarrow \)(iv) Obvious.

(iv)\(\Rightarrow \)(ii) Let \(\{X_n\}_{n=1}^\infty \) be as in Corollary 4.6 (with \(m=1\)).

Then, by (3.4), we see that \(\chi _{X_n}, \chi _{\mathsf{N }_\phi \cap X_n} \in \mathcal{D }(C_\phi )\) and \(\Vert C_\phi (\chi _{\mathsf{N }_\phi \cap X_n})\Vert ^2 = \int _{\mathsf{N }_\phi \cap X_n} \mathsf{h }_\phi \mathrm{d}\mu = 0\) for all \(n \in \mathbb N \), which together with our assumption that \(\mathcal N (C_\phi ) \subseteq \mathcal N (C_\phi ^*)\) yields

$$\begin{aligned} 0 = \langle \chi _{\mathsf{N }_\phi \cap X_n},C_\phi \chi _{X_n}\rangle = \int \limits _{\mathsf{N }_\phi \cap X_n}\chi _{X_n} \circ \phi \mathrm{d}\mu = \mu (\mathsf{N }_\phi \cap X_n \cap \phi ^{-1}(X_n)) \end{aligned}$$

for all \(n \in \mathbb N \). Since \(\mathsf{N }_\phi \cap X_n \cap \phi ^{-1}(X_n) \nearrow \mathsf{N }_\phi \) as \(n\rightarrow \infty \), the continuity of measure implies that \(\mu (\mathsf{N }_\phi )=0\). This completes the proof. \(\square \)

Corollary 6.3

If \(C_\phi \) is hyponormal, then \(\mathcal N (C_\phi ) = \{0\}\).

Proof

It follows from the definition of hyponormality that \(\mathcal N (C_\phi ) \subseteq \mathcal N (C_\phi ^*)\). This and Proposition 6.2 complete the proof. \(\square \)

Corollary 6.4

If \(C_\phi \) is formally normal, then

$$\begin{aligned} \mathcal{D }(C_\phi ) \cap \mathcal N (C_\phi ^*) = \{0\}. \end{aligned}$$

Proof

If \(f \in \mathcal{D }(C_\phi ) \cap \mathcal N (C_\phi ^*)\), then \(\Vert C_\phi f\Vert = \Vert C_\phi ^* f\Vert =0\), which means that \(f \in \mathcal N (C_\phi )\). Applying Corollary 6.3 completes the proof. \(\square \)

It turns out that composition of \(\mathsf{h }_\phi \) with \(\phi \) is positive a.e. \([\mu ]\) (see also the proof of [23, Corollary 5]).

Proposition 6.5

If \(\phi :X \rightarrow X\) is nonsingular, then \(\mathsf{h }_\phi \circ \phi > 0\) a.e. \([\mu ]\).

Proof

Note that \(\mu (\phi ^{-1}(\mathsf{N }_\phi )) = \int _X \chi _{\mathsf{N }_\phi } \circ \phi \mathrm{d}\mu = \int _X \chi _{\mathsf{N }_\phi } \mathsf{h }_\phi \mathrm{d}\mu = 0\). This combined with \(\phi ^{-1}(\mathsf{N }_\phi ) = \{x \in X :\mathsf{h }_\phi (\phi (x))=0\}\) completes the proof. \(\square \)

Corollary 6.6

If \(\phi \) is a nonsingular transformation of \(X\) and \(\mathsf{h }_\phi \circ \phi = \mathsf{h }_\phi \) a.e. \([\mu ]\), then \(\mathcal N (C_\phi ) = \{0\}\).

Proof

Apply Propositions 6.1 and 6.5. \(\square \)

7 The polar decomposition

Given an \({\fancyscript{A}}\)-measurable function \(u:X \rightarrow \mathbb{C }\), we denote by \(M_u\) the operator of multiplication by \(u\) in \(L^2(\mu )\) defined by

$$\begin{aligned} \mathcal{D }(M_u)&= \{f \in L^2(\mu ):u\cdot f \in L^2(\mu )\},\\ M_u f&= u \cdot f, \quad f \in \mathcal{D }(M_u). \end{aligned}$$

The operator \(M_u\) is a normal operator (cf. [3, Section 7.2].

The polar decomposition of \(C_{\phi }\) can be explicitly described as follows.

Proposition 7.1

Suppose that the composition operator \(C_{\phi }\) is densely defined and \(C_{\phi }=U|C_{\phi }|\) is its polar decomposition. Then,

  1. (i)

    \(|C_\phi |=M_{\mathsf{h }_\phi ^{1/2}}\),

  2. (ii)

    the initial space of \(U\) is given by Footnote 3

    $$\begin{aligned} \overline{\mathcal{R }(|C_{\phi }|)} = \left\{ \mathsf{h }_\phi ^{1/2} f :f \in L^2(\mathsf{h }_\phi \mathrm{d}\mu )\right\} , \end{aligned}$$
    (7.1)
  3. (iii)

    the final space of \(U\) is given by

    $$\begin{aligned} \overline{\mathcal{R }(C_{\phi })} = \big \{f\circ \phi :f \in L^2(\mathsf{h }_\phi \mathrm{d}\mu )\big \}, \end{aligned}$$
    (7.2)
  4. (iv)

    the partial isometry \(U\) is given by Footnote 4

    $$\begin{aligned} Ug = \frac{g \circ \phi }{(\mathsf{h }_\phi \circ \phi )^{1/2}}, \quad g\in L^2(\mu ), \end{aligned}$$
    (7.3)
  5. (v)

    the adjoint \(U^*\) of \(U\) is given by

    $$\begin{aligned} U^* g = \mathsf{h }_\phi ^{1/2} \cdot V^{-1}Pg, \quad g \in L^2(\mu ), \end{aligned}$$

    where \(V:L^2(\mathsf{h }_\phi \mathrm{d}\mu ) \rightarrow \overline{\mathcal{R }(C_{\phi })}\) is a unitary operator defined by \(Vf = f\circ \phi \) for \(f\in L^2(\mathsf{h }_\phi \mathrm{d}\mu )\) and \(P\) is the orthogonal projection of \(L^2(\mu )\) onto \(\overline{\mathcal{R }(C_{\phi })}\).

Proof

  1. (i)

    We will show that \(C_{\phi }^* C_{\phi } \subseteq M_{\mathsf{h }_\phi }\). Let \(\{X_n\}_{n=1}^\infty \) be as in Corollary 4.6 (with \(m=1\)). Take \(f \in \mathcal{D }(C_{\phi }^* C_{\phi })\) and fix \(n\in \mathbb N \). By (3.5), \(\chi _{\varDelta } \in \mathcal{D }(C_{\phi })\) whenever \(\varDelta \in {\fancyscript{A}}\) and \(\varDelta \subseteq X_n\). Thus, for every such \(\varDelta \), we have

    $$\begin{aligned} \int \limits _{\varDelta } C_{\phi }^* C_{\phi }f \mathrm{d}\mu = \langle C_{\phi }^* C_{\phi }f,\chi _{\varDelta }\rangle = \langle C_{\phi }f,C_{\phi }\chi _{\varDelta }\rangle \overset{(3.2)}{=} \int \limits _{\varDelta } f \mathsf{h }_\phi \mathrm{d}\mu . \end{aligned}$$

    Since both functions \((C_{\phi }^* C_{\phi }f)\chi _{X_n}\) and \((f \mathsf{h }_\phi ) \chi _{X_n}\) are in \(L^1(\mu )\), we deduce that \(C_{\phi }^* C_{\phi }f = f \mathsf{h }_\phi \) a.e. \([\mu ]\) on \(X_n\). This and \(X_n\nearrow X\) give \(C_{\phi }^* C_{\phi }f = f \mathsf{h }_\phi \) a.e. \([\mu ]\). As a consequence, we have \(C_{\phi }^* C_{\phi } \subseteq M_{\mathsf{h }_\phi }\). Since both are selfadjoint operators, they are equal. Thus \(|C_{\phi }| = M_{\mathsf{h }_\phi }^{1/2} = M_{\mathsf{h }_\phi ^{1/2}}\).

  2. (ii)

    By [3, Section 8.1] and Proposition 6.1, we have

    $$\begin{aligned} \overline{\mathcal{R }(|C_{\phi }|)} = \mathcal N (|C_{\phi }|)^\perp =\mathcal N (C_{\phi })^\perp = \chi _{X \setminus \mathsf{N }_{\phi }} L^2(\mu ), \end{aligned}$$
    (7.4)

    which as easily seen gives (7.1).

  3. (iii)

    By (3.4) and (ii), the mapping \(W:\overline{\mathcal{R }(|C_{\phi }|)} \rightarrow L^2(\mu )\) given by

    $$\begin{aligned} W(\mathsf{h }_\phi ^{1/2} f) = f \circ \phi , \quad f \in L^2(\mathsf{h }_\phi \mathrm{d}\mu ), \end{aligned}$$
    (7.5)

    is a well-defined isometry. Using (i), we verify that \(W|_{\mathcal{R }(|C_{\phi }|)} = U|_{\mathcal{R }(|C_{\phi }|)}\), which implies that \(\overline{\mathcal{R }(C_{\phi })} = \mathcal{R }(U)=\mathcal{R }(W)\). Hence, (iii) holds and, by (7.5), we have

    $$\begin{aligned} U^*(f\circ \phi ) = \mathsf{h }_\phi ^{1/2} f, \quad f \in L^2(\mathsf{h }_\phi \mathrm{d}\mu ). \end{aligned}$$
    (7.6)
  4. (iv)

    Applying the measure transport theorem to the restriction of \(\phi \) to the full \(\mu \)-measure set on which \(\mathsf{h }_\phi \circ \phi \) is positive (cf. Proposition 6.5), we get

    $$\begin{aligned} \int \limits _X \frac{|g\circ \phi |^2}{\mathsf{h }_\phi \circ \phi } \mathrm{d}\mu = \int \limits _{X \setminus \mathsf{N }_\phi } |g|^2 \mathrm{d}\mu , \quad g \in L^2(\mu ). \end{aligned}$$
    (7.7)

    This and Proposition 6.1 imply that the mapping \(\tilde{U} :L^2(\mu ) \ni g \mapsto \frac{g \circ \phi }{(\mathsf{h }_\phi \circ \phi )^{1/2}} \in L^2(\mu )\) is a contraction such that \(\mathcal N (\tilde{U})=\chi _{\mathsf{N }_\phi }L^2(\mu ) = \mathcal N (|C_\phi |)\). Hence, by (7.4) and (7.7), \(\tilde{U}\) is an isometry on \(\overline{\mathcal{R }(|C_\phi |)}\). Clearly, by (i), \(\tilde{U}|C_\phi |g = C_\phi g\) for \(g \in \mathcal{D }(C_\phi )=\mathcal{D }(|C_\phi |)\), which implies that \(U=\tilde{U}\).

  5. (v)

    By (3.4) and (7.2), \(V\) is a well-defined unitary operator. If \(g \in L^2(\mu )\), then by (iii), \(Pg=f\circ \phi \) a.e. \([\mu ]\) for some \(f \in L^2(\mathsf{h }_\phi \mathrm{d}\mu )\). Thus, by \(\mathcal N (U^*)=\mathcal{R }(I-P)\) and (7.6), we have

    $$\begin{aligned} U^*g = U^*P g = U^*(f\circ \phi ) = \mathsf{h }_\phi ^{1/2} f = \mathsf{h }_\phi ^{1/2} \cdot V^{-1} Pg. \end{aligned}$$

    This completes the proof.

\(\square \)

Regarding Proposition 7.1, we note that the formulas for \(|C_\phi |\) and \(\overline{\mathcal{R }(C_\phi )}\) are well-known in the case of bounded composition operators (cf. [23, Lemma 1]). The formula (7.3) has appeared in [10, p. 387] in the context of bounded operators without proof.

Corollary 7.2

Suppose that \(C_{\phi }\) is densely defined and \(g \in L^2(\mu )\). Then \(g\) belongs to \(\overline{\mathcal{R }(C_{\phi })}\) if and only if one of the following equivalent conditions holdsFootnote 5 :

  1. (i)

    there is an \({\fancyscript{A}}\)-measurable function \(f:X \rightarrow \mathbb{C }\) such that \(g=f\circ \phi \) a.e. \([\mu ]\),

  2. (ii)

    there is a \(\phi ^{-1}({\fancyscript{A}})\)-measurable function \(f:X \rightarrow \mathbb{C }\) such that \(g=f\) a.e. \([\mu ]\),

  3. (iii)

    \(g\) is \((\phi ^{-1}({\fancyscript{A}}))^\mu \)-measurable,

  4. (iv)

    for every Borel set \(\varDelta \) in \(\mathbb{C }\) there exists \(\varDelta ^\prime \in {\fancyscript{A}}\) such that

    $$\begin{aligned} \mu \big (g^{-1}(\varDelta ) \vartriangle \phi ^{-1}(\varDelta ^\prime )\big )=0. \end{aligned}$$

In particular, \(\overline{\mathcal{R }(C_{\phi })} = L^2(\mu |_{(\phi ^{-1}({\fancyscript{A}}))^\mu })\).

Proof

Apply (3.4), (7.2), (13.1), (13.2) and Lemma 13.3. \(\square \)

Corollary 7.3

If \(C_{\phi }\) is densely defined, then the map \(V:L^2(\mathsf{h }_\phi \mathrm{d}\mu ) \rightarrow \overline{\mathcal{R }(C_{\phi })}\) given by \(Vf = f\circ \phi \) for \(f\in L^2(\mathsf{h }_\phi \mathrm{d}\mu )\) is a well-defined unitary operator such that

$$\begin{aligned} \mathcal{D }(C_{\phi }^*)&= \big \{g \in L^2(\mu ):\mathsf{h }_\phi \cdot V^{-1}Pg \in L^2(\mu )\big \},\nonumber \\ C_{\phi }^*g&= \mathsf{h }_\phi \cdot V^{-1}Pg, \quad g \in \mathcal{D }(C_{\phi }^*), \end{aligned}$$
(7.8)

where \(P\) is the orthogonal projection of \(L^2(\mu )\) onto \(\overline{\mathcal{R }(C_{\phi })}=L^2(\mu |_{(\phi ^{-1}({\fancyscript{A}}))^\mu })\).

Proof

If \(C_{\phi }=U|C_{\phi }|\) is the polar decomposition of \(C_{\phi }\), then \(C_{\phi }^*=|C_{\phi }|U^*\). This, Proposition 7.1 and Corollary 7.2 complete the proof. \(\square \)

Remark 7.4

Concerning Corollary 7.3, we observe that, in view of (13.3), \(\mathsf{E }(g):=\mathsf{E }(g|\phi ^{-1}({\fancyscript{A}}))=Pg\) a.e. \([\mu ]\) and thus \(C_\phi ^* g = \mathsf{h }_\phi \cdot (\mathsf{E }(g)\circ \phi ^{-1})\) for every \(g\in \mathcal{D }(C_\phi ^*)\), where \(\mathsf{E }(g)\circ \phi ^{-1}\) is understood as in [11, Lemma 6.4].

8 Normality and quasinormality

It turns out that the characterizations of quasinormality and normality of unbounded composition operators take the same forms as those for bounded ones.

Proposition 8.1

If \(C_{\phi }\) is densely defined, then \(C_{\phi }\) is quasinormal if and only if \(\mathsf{h }_\phi = \mathsf{h }_\phi \circ \phi \) a.e. \([\mu ]\).

Proof

Let \(C_\phi = U|C_\phi |\) be the polar decomposition of \(C_\phi \). Suppose that \(C_\phi \) is quasinormal. Then, by [49, Proposition 1], \(U |C_\phi | \subseteq |C_\phi | U\). Let \(\{X_n\}_{n=1}^\infty \) be as in Corollary 4.6 (with \(m=1\)). Then, by (3.5), \(\{\chi _{X_n}\}_{n=1}^\infty \subseteq \mathcal{D }(C_\phi )\), which together with Proposition 7.1 implies that for every \(n\in \mathbb N \),

$$\begin{aligned} \chi _{X_n} \circ \phi = U |C_\phi | \chi _{X_n} = |C_\phi | U \chi _{X_n} = \Big (\frac{\mathsf{h }_\phi }{\mathsf{h }_\phi \circ \phi }\Big )^{1/2} \, \chi _{X_n} \circ \phi \quad \text{ a.e.} [\mu ]\text{.} \end{aligned}$$

Since \(X_n \nearrow X\) as \(n\rightarrow \infty \), we conclude that \(\mathsf{h }_\phi = \mathsf{h }_\phi \circ \phi \) a.e. \([\mu ]\).

For the converse, take \(f \in \mathcal{D }(|C_\phi |)\). By (7.3) and \(\mathcal{D }(|C_\phi |)=\mathcal{D }(C_\phi )\), we have

$$\begin{aligned} \mathsf{h }_\phi ^{1/2} Uf = \Big (\frac{\mathsf{h }_\phi }{\mathsf{h }_\phi \circ \phi }\Big )^{1/2} f\circ \phi = f\circ \phi \in L^2(\mu ). \end{aligned}$$

Hence, by Proposition 7.1 (i), \(f \in \mathcal{D }(|C_\phi |U)\) and \(|C_\phi |U f = C_\phi f = U|C_\phi |f\). Therefore, \(U|C_\phi | \subseteq |C_\phi |U\). Applying [49, Proposition 1] completes the proof. \(\square \)

Proposition 8.2

If \(\overline{\mathcal{D }(C_{\phi })}=L^2(\mu )\), then the following are equivalent:

  1. (i)

    \(C_{\phi }\) is normal,

  2. (ii)

    \(\mathsf{h }_\phi = \mathsf{h }_\phi \circ \phi \) a.e. \([\mu ]\) and \(\mathcal N (C_{\phi }^*) \subseteq \mathcal N (C_{\phi })\),

  3. (iii)

    \(\mathsf{h }_\phi = \mathsf{h }_\phi \circ \phi \) a.e. \([\mu ]\) and \(\mathcal N (C_{\phi }^*)=\{0\}\),

  4. (iv)

    \(\mathsf{h }_\phi = \mathsf{h }_\phi \circ \phi \) a.e. \([\mu ]\) and for every \(\varDelta \in {\fancyscript{A}}\) there exists \(\varDelta ^\prime \in {\fancyscript{A}}\) such that \(\mu (\varDelta \vartriangle \phi ^{-1}(\varDelta ^\prime ))=0\).

Moreover, if \(C_{\phi }\) is normal, then \(\mathcal N (C_{\phi })=\{0\}\) and \(\mathsf{h }_\phi > 0\) a.e. \([\mu ]\).

Proof

(i) \(\Rightarrow \)(iii) Since normal operators are always quasinormal, we infer from Proposition 8.1 that \(\mathsf{h }_\phi = \mathsf{h }_\phi \circ \phi \) a.e. \([\mu ]\). Clearly, \(\mathcal N (C_{\phi }) = \mathcal N (C_{\phi }^*)\). That \(\mathcal N (C_{\phi }^*)=\{0\}\) follows from Corollary 6.6.

(iii)\(\Rightarrow \)(ii) Evident.

(ii)\(\Rightarrow \)(i) This is a direct consequence of Proposition 8.1 and Theorem 2.2.

(iii)\(\Leftrightarrow \)(iv) Since \(\mathcal N (C_{\phi }^*) =\{0\}\) if and only if \(\mathcal{R }(C_{\phi })\) is dense in \(L^2(\mu )\), it suffices to apply Corollary 7.2, Lemma 13.2 and (13.1).

The ”moreover” part follows from the above and Proposition 6.5. \(\square \)

9 Formal normality

In this section, we show that formally normal composition operators are normal. The proof of this result relies on a characterization of composition operators for which \(\mathsf{h }_{\phi ^2} = \mathsf{h }_{\phi }^2\) a.e. \([\mu ]\). We also provide an alternative proof which depends heavily on the fact that quasinormal formally normal operators are normal.

We begin by proving a result which is of measure-theoretic nature. We refer the reader to “Appendix B” for the definition and basic properties of \(\mathsf{E }(\cdot |\phi ^{-1}({\fancyscript{A}}))\) (which makes sense if \(\mathsf{h }_{\phi } < \infty \) a.e. \([\mu ]\)). For brevity, we write \(\mathsf{E }(\cdot ) = \mathsf{E }(\cdot |\phi ^{-1}({\fancyscript{A}}))\).

Lemma 9.1

If \(\phi \) is a nonsingular transformation of \(X\) such that \(\mathsf{h }_{\phi } < \infty \) a.e. \([\mu ]\), then the following two conditions are equivalent for every \(n\in \mathbb N \) :

  1. (i)

    \(\mathsf{h }_{\phi ^{n+1}} = \mathsf{h }_{\phi ^n} \cdot \mathsf{h }_\phi \) a.e. \([\mu ]\),

  2. (ii)

    \(\mathsf{E }(\mathsf{h }_{\phi ^n}) = \mathsf{h }_{\phi ^n} \circ \phi \) a.e. \([\mu |_{\phi ^{-1}({\fancyscript{A}})}]\).

Proof

(i)\(\Rightarrow \)(ii) Note that

$$\begin{aligned} \int _{\phi ^{-1}(\varDelta )} \mathsf{E }(\mathsf{h }_{\phi ^n}) \mathrm{d}\mu&= \int _{\phi ^{-1}(\varDelta )} \mathsf{h }_{\phi ^n} \mathrm{d}\mu = \mu ((\phi ^{-n}(\phi ^{-1}(\varDelta ))))\\&= \mu ((\phi ^{-(n+1)}(\varDelta ))) = \int _{\varDelta } \mathsf{h }_{\phi ^{n+1}} \mathrm{d}\mu = \int _{\varDelta } \mathsf{h }_{\phi ^n} \cdot \mathsf{h }_\phi \mathrm{d}\mu \\&= \int _X (\chi _\varDelta \circ \phi ) (\mathsf{h }_{\phi ^n} \circ \phi ) \mathrm{d}\mu = \int \limits _{\phi ^{-1}(\varDelta )} \mathsf{h }_{\phi ^n} \circ \phi \mathrm{d}\mu , \quad \varDelta \in {\fancyscript{A}}, \end{aligned}$$

which, by the uniqueness assertion in the Radon-Nikodym theorem, implies (ii).

Arguing as above, we can prove the reverse implication. \(\square \)

The next two lemmas are key ingredients of the proof of Theorem 9.4. The first lemma shows that \(\mathsf{h }_{\phi ^2} = \mathsf{h }_{\phi }^2\) a.e. \([\mu ]\) if and only if \(C_{\phi }\) is “formally normal” on its range. This result is of some independent interest.

Lemma 9.2

Suppose that \(C_\phi \) is densely defined. Then, the following two conditions are equivalent:

  1. (i)

    \(C_\phi ^2\) is densely defined, \(\mathcal{D }(C_\phi ^2) \subseteq \mathcal{D }(C_\phi ^* C_\phi )\) and \(\Vert C_\phi ^2 f\Vert = \Vert C_\phi ^*C_\phi f\Vert \) for every \(f\in \mathcal{D }(C_\phi ^2)\),

  2. (ii)

    \(\mathsf{h }_{\phi ^2} = \mathsf{h }_{\phi }^2\) a.e. \([\mu ]\).

Proof

(i)\(\Rightarrow \)(ii) Take \(f \in \mathcal{D }(C_\phi ^2)\). Then, by Proposition 7.1(i), we have

$$\begin{aligned} \int \limits _X |f|^2 \mathsf{h }_\phi ^2 \mathrm{d}\mu = \Vert M_{\mathsf{h }_\phi } f\Vert ^2 = \Vert C_\phi ^* C_\phi f\Vert ^2 = \Vert C_{\phi ^2} f\Vert ^2 = \int \limits _X |f|^2 \mathsf{h }_{\phi ^2} \mathrm{d}\mu . \end{aligned}$$
(9.1)

Let \(\{X_n\}_{n=1}^\infty \) be as in Corollary 4.6 (with \(m=2\)). Then, \(\{\chi _{X_n}\}_{n=1}^\infty \subseteq \mathcal{D }(C_\phi ^2)\) and

$$\begin{aligned} \int \limits _{\varDelta } \mathsf{h }_\phi ^2 \mathrm{d}\mu \overset{(9.1)}{=} \int \limits _{\varDelta } \mathsf{h }_{\phi ^2} \mathrm{d}\mu < \infty , \quad \varDelta \in {\fancyscript{A}}, \, \varDelta \subseteq X_n, \, n \in \mathbb N , \end{aligned}$$

which implies that \(\mathsf{h }_\phi ^2 = \mathsf{h }_{\phi ^2}\) a.e. \([\mu ]\) on \(X_n\) for every \(n\in \mathbb N \). Hence, (ii) holds.

(ii)\(\Rightarrow \)(i) Since, by Proposition 3.2, \(\mathsf{h }_{\phi } < \infty \) a.e. \([\mu ]\), we infer from (ii) and Corollary 4.5 that \(C_{\phi }^2\) is densely defined. Now we take \(f\in \mathcal{D }(C_\phi ^2)\). Then, by (3.6), \(\int _X |f \mathsf{h }_\phi |^2 \mathrm{d}\mu = \int _X |f|^2 \mathsf{h }_{\phi ^2} \mathrm{d}\mu < \infty \), which means that \(f \in \mathcal{D }(M_{\mathsf{h }_\phi }) = \mathcal{D }(C_\phi ^*C_\phi )\). Arguing as in (9.1), we obtain (i). \(\square \)

It is worth pointing out that implication (i)\(\Rightarrow \)(ii) of Lemma 9.2 is no longer true if we drop the assumption that \(C_\phi ^2\) is densely defined. To see this, it is enough to consider a nonzero densely defined composition operator whose square has trivial domain and to apply Corollary 4.5. For examples of such operators, we refer the reader to recent articles [26] and [6].

Lemma 9.3

If \(\phi \) is nonsingular transformation of \(X\), then the following conditions are equivalent:

  1. (i)

    \(C_\phi \) is normal,

  2. (ii)

    \(C_\phi \) is formally normal and \(\overline{\mathcal{D }(C_\phi ^2)}=L^2(\mu )\).

Proof

(i)\(\Rightarrow \)(ii) Evident (since powers of normal operators are normal, cf. [3]).

(ii)\(\Rightarrow \)(i) First we will show that

$$\begin{aligned} \mathcal{D }(C_\phi ) \cap \overline{\mathcal{R }(C_\phi )} = \mathcal{D }(C_\phi ^*) \cap \overline{\mathcal{R }(C_\phi )}. \end{aligned}$$
(9.2)

Indeed, if \(g \in \mathcal{D }(C_\phi ^*) \cap \overline{\mathcal{R }(C_\phi )}\), then by (7.2) there exists \(f \in L^2(\mathsf{h }_\phi \mathrm{d}\mu )\) such that \(g = f \circ \phi \) a.e. \([\mu ]\). It follows from Corollary 7.3 that \(\mathsf{h }_\phi f = \mathsf{h }_\phi V^{-1}g \in L^2(\mu )\). This combined with the fact that (ii) implies condition (i) of Lemma 9.2 leads to

$$\begin{aligned} \int \limits _X |g\circ \phi |^2 \mathrm{d}\mu = \int \limits _X |f \circ \phi ^2|^2 \mathrm{d}\mu = \int \limits _X |f|^2 \mathsf{h }_{\phi ^2}\mathrm{d}\mu = \int \limits _X |f\mathsf{h }_{\phi }|^2\mathrm{d}\mu < \infty , \end{aligned}$$

which means that \(g \in \mathcal{D }(C_\phi )\). This yields (9.2).

Let \(P\) be the orthogonal projection of \(L^2(\mu )\) onto \(\overline{\mathcal{R }(C_{\phi })}\). We will prove that

$$\begin{aligned} P \mathcal{D }(C_\phi ) \subseteq \mathcal{D }(C_\phi ). \end{aligned}$$
(9.3)

Indeed, take \(f \in \mathcal{D }(C_\phi )\). Since \((I-P)f \in \mathcal N (C_\phi ^*)\) and \(\mathcal{D }(C_\phi ) \subseteq \mathcal{D }(C_\phi ^*)\), we get \(Pf \in \mathcal{D }(C_\phi ^*) \cap \overline{\mathcal{R }(C_\phi )}\). Hence, by (9.2), \(Pf\in \mathcal{D }(C_\phi )\), which proves (9.3).

It follows from (9.3) and Corollary 6.4 that

$$\begin{aligned} \mathcal{D }(C_\phi ) \subseteq \big (\mathcal{D }(C_\phi )\cap \mathcal N (C_\phi ^*)\big ) \oplus \big (\mathcal{D }(C_\phi )\cap \overline{\mathcal{R }(C_\phi )}\big ) = \mathcal{D }(C_\phi )\cap \overline{\mathcal{R }(C_\phi )}, \end{aligned}$$

which together with \(\overline{\mathcal{D }(C_\phi )}=L^2(\mu )\) imply that \(\overline{\mathcal{R }(C_\phi )}=L^2(\mu )\). Therefore, by (9.2), \(\mathcal{D }(C_\phi )=\mathcal{D }(C_\phi ^*)\), which completes the proof. \(\square \)

Now we show that the assumption \(\overline{\mathcal{D }(C_\phi ^2)}=L^2(\mu )\) can be dropped without affecting the conclusion of Lemma 9.3.

Theorem 9.4

Let \(\phi \) be a nonsingular transformation of \(X\). Then, \(C_\phi \) is normal if and only if \(C_\phi \) is formally normal.

Proof

It suffices to prove the “if” part. Suppose \(C_\phi \) is formally normal. Let \(\{X_n\}_{n=1}^\infty \subseteq {\fancyscript{A}}\) be as in Corollary 4.6 (with \(m=1\)). Take \(\varDelta \in {\fancyscript{A}}\). Since \(\{\chi _{X_n\cap \varDelta }\}_{n=1}^\infty \subseteq \mathcal{D }(C_\phi )\), we get (see also Remark 7.4)

$$\begin{aligned} \int \limits _{X_n\cap \varDelta } \mathsf{h }_\phi \mathrm{d}\mu&\overset{(3.4)}{=} \Vert C_\phi (\chi _{X_n\cap \varDelta })\Vert ^2 = \Vert C_\phi ^* (\chi _{X_n\cap \varDelta })\Vert ^2\\&\overset{(7.8)}{=} \int \limits _X \mathsf{h }_\phi ^2 \cdot |V^{-1}\mathsf{E }(\chi _{X_n\cap \varDelta })|^2 \mathrm{d}\mu \\& = \int \limits _X (\mathsf{h }_\phi \circ \phi )(\mathsf{E }(\chi _{X_n\cap \varDelta }))^2 \mathrm{d}\mu , \quad n \in \mathbb N . \end{aligned}$$

Using (13.6) and Lebesgue’s monotone convergence theorem, we obtain

$$\begin{aligned} \int \limits _{\varDelta } \mathsf{h }_\phi \mathrm{d}\mu = \int \limits _X (\mathsf{h }_\phi \circ \phi )(\mathsf{E }(\chi _{\varDelta }))^2 \mathrm{d}\mu , \quad \varDelta \in {\fancyscript{A}}, \end{aligned}$$

which yields

$$\begin{aligned} \int \limits _{\phi ^{-1}(\varDelta )} \mathsf{h }_\phi \mathrm{d}\mu = \int \limits _{\phi ^{-1}(\varDelta )} \mathsf{h }_\phi \circ \phi \mathrm{d}\mu , \quad \varDelta \in {\fancyscript{A}}. \end{aligned}$$

This in turn implies that \(\mathsf{E }(\mathsf{h }_\phi )=\mathsf{h }_\phi \circ \phi \) a.e. \([\mu |_{\phi ^{-1}({\fancyscript{A}})}]\). By Lemma 9.1, \(\mathsf{h }_{\phi ^{2}} = \mathsf{h }_{\phi }^2\) a.e. \([\mu ]\). Since \(\mathsf{h }_\phi < \infty \) a.e. \([\mu ]\), we see that \(\mathsf{h }_\phi + \mathsf{h }_{\phi ^{2}} < \infty \) a.e. \([\mu ]\). Using Corollary 4.5 (ii), we get \(\overline{\mathcal{D }(C_\phi ^2)}=L^2(\mu )\). Applying Lemma 9.3 completes the proof. \(\square \)

Remark 9.5

The “if” part of Theorem 9.4 can be proved using a different more advanced way. Indeed, assume that \(C_\phi \) is formally normal. Then, by the polarization formula, we have

$$\begin{aligned} \int \limits _X f \bar{g} \mathsf{h }_\phi \mathrm{d}\mu&= \langle C_\phi f,C_\phi g\rangle = \langle C_\phi ^* f,C_\phi ^* g\rangle \nonumber \\&\overset{(7.8)}{=} \int _X \mathsf{h }_\phi ^2 \big (V^{-1}\mathsf{E }(f)\big ) \big (\overline{V^{-1}\mathsf{E }(g)}\big ) \mathrm{d}\mu \nonumber \\&= \int _X (\mathsf{h }_\phi \circ \phi ) \, \mathsf{E }(f) \, \overline{\mathsf{E }(g)} \mathrm{d}\mu , \quad f,g\in \mathcal{D }(C_\phi ). \end{aligned}$$
(9.4)

By Propositions 3.2 and 6.1, and Corollary 6.3, we can assume that \(0 < \mathsf{h }_\phi (x) < \infty \) for all \(x\in X\). Let \(\{X_n\}_{n=1}^\infty \subseteq {\fancyscript{A}}\) be as in Corollary 4.6 (with \(m=1\)). Set \(Y_n=\{x \in X_n:\mathsf{h }_\phi (x) {\geqslant }1/n\}\) for \(n\in \mathbb N \). Clearly, \(Y_n \nearrow X\) as \(n\rightarrow \infty \). Take \(\varDelta \in {\fancyscript{A}}\). Since \(\{\chi _{Y_n}\}_{n=1}^\infty , \left\{ \mathsf{h }_\phi ^{-1}\cdot \chi _{Y_n \cap \varDelta } \right\} _{n=1}^\infty \subseteq \mathcal{D }(C_\phi )\), we can substitute \(f=\mathsf{h }_\phi ^{-1}\cdot \chi _{Y_n \cap \varDelta }\) and \(g=\chi _{Y_n}\) into (9.4). What we get is

$$\begin{aligned} \mu (Y_n \cap \varDelta )&= \int _X (\mathsf{h }_\phi \circ \phi ) \, \mathsf{E }(\mathsf{h }_\phi ^{-1}\cdot \chi _{Y_n \cap \varDelta }) \, \mathsf{E }(\chi _{Y_n}) \mathrm{d}\mu \\&\overset{(13.5)}{=} \int _{Y_n \cap \varDelta } \frac{\mathsf{h }_\phi \circ \phi }{\mathsf{h }_\phi } \, \mathsf{E }(\chi _{Y_n}) \mathrm{d}\mu , \quad n \in \mathbb N . \end{aligned}$$

Using (13.6) and Lebesgue’s monotone convergence theorem, we obtain

$$\begin{aligned} \mu (\varDelta ) = \int \limits _{\varDelta } \frac{\mathsf{h }_\phi \circ \phi }{\mathsf{h }_\phi } \mathrm{d}\mu , \quad \varDelta \in {\fancyscript{A}}, \end{aligned}$$

which implies that \(\mathsf{h }_\phi \circ \phi = \mathsf{h }_\phi \) a.e. \([\mu ]\). By Proposition 8.1, \(C_\phi \) is quasinormal. Since quasinormal formally normal operators are normal (cf. [49, Corollary 4]; see also [52]), the proof is complete.

10 Generating Stieltjes moment sequences

We begin by proving two lemmas which are main tools in the proof of Theorem 10.4 below.

Lemma 10.1

Suppose \(\phi \) is a nonsingular transformation of \(X\) and \(\left\{ \mathcal{E }_n \right\} _{n=1}^\infty \) is a sequence of subsets of \(L^2(\mu )\) satisfying the following three conditions:

  1. (i)

    \(\mathcal{E }_n\) fulfills (12.5), \(\mathcal{E }_n \subseteq \mathcal{D }(C_\phi ^n)\) and \(\overline{\mathcal{E }_n}=L^2(\mu )\) for all \(n\in \mathbb N \),

  2. (ii)

    \(\big [\Vert C_\phi ^{i+j}f\Vert ^2\big ]_{i,j=0}^n {\geqslant }0\) for all \(f\in \mathcal{E }_{2n}\) and \(n\in \mathbb N \),

  3. (iii)

    \(\big [\Vert C_\phi ^{i+j+1}f\Vert ^2\big ]_{i,j=0}^n {\geqslant }0\) for all \(f\in \mathcal{E }_{2n+1}\) and \(n\in \mathbb N \).

Then the following three assertions hold:

  1. (a)

    \( \left\{ \mathsf{h }_{\phi ^n}(x) \right\} _{n=0}^\infty \) is a Stieltjes moment sequence for \(\mu \)-a.e. \(x \in X\),

  2. (b)

    \(C_\phi ^n=C_{\phi ^n}\) for every \(n\in \mathbb N \),

  3. (c)

    \(\mathcal{D }^\infty (C_\phi )\) is a core for \(C_\phi ^n\) for every \(n\in \mathbb{Z }_+\).

Proof

(a) By (i) and Corollary 4.5 (ii), there is no loss of generality in assuming that \(0{\leqslant }\mathsf{h }_{\phi ^n} (x) < \infty \) for all \(x \in X\) and \(n\in \mathbb{Z }_+\). Using (3.4), we obtain

$$\begin{aligned} \int \limits _X\Big |\sum _{i,j=0}^n \alpha _i\bar{\alpha }_j \mathsf{h }_{\phi ^{i+j}}\Big ||f|^2 \mathrm{d}\mu < \infty , \quad f \in \mathcal{D }(C_\phi ^{2n}), \, \{\alpha _i\}_{i=0}^n \subseteq \mathbb{C }, \, n\in \mathbb{Z }_+. \end{aligned}$$
(10.1)

If \(\{\alpha _i\}_{i=0}^n \subseteq \mathbb{C }\), then by (i) and (ii) we have

$$\begin{aligned} 0 {\leqslant }\sum _{i,j=0}^n \Vert C_\phi ^{i+j} f\Vert ^2 \alpha _i \bar{\alpha }_j = \int \limits _X \left(\sum _{i,j=0}^n \alpha _i\bar{\alpha }_j \mathsf{h }_{\phi ^{i+j}}\right)|f|^2 \mathrm{d}\mu , \quad f \in \mathcal{E }_{2n}, \, n\in \mathbb N . \end{aligned}$$

Combining (i), (10.1) and Corollary 12.6 (with \(\mathcal{E }=\mathcal{E }_{2n}\)), we see that

$$\begin{aligned} \sum _{i,j=0}^n \alpha _i\bar{\alpha }_j \mathsf{h }_{\phi ^{i+j}} {\geqslant }0 \; \text{ a.e.}\,\, [\mu ]\; \text{ for} \text{ all}\; n\in \mathbb N \; \text{ and}\; \{\alpha _i\}_{i=0}^n \subseteq \mathbb{C }. \end{aligned}$$

Let \(Q\) be a countable dense subset of \(\mathbb{C }\). Then, there exists a set \(\varDelta _0 \in {\fancyscript{A}}\) such that \(\mu (X\setminus \varDelta _0)=0\) and \(\sum _{i,j=0}^n \alpha _i\bar{\alpha }_j \mathsf{h }_{\phi ^{i+j}}(x) {\geqslant }0\) for all \(n\in \mathbb N , \{\alpha _i\}_{i=0}^n \subseteq Q\) and \(x \in \varDelta _0\). As \(Q\) is dense in \(\mathbb{C }\), we conclude that \([\mathsf{h }_{\phi ^{i+j}}(x)]_{i,j=0}^n {\geqslant }0\) for all \(n\in \mathbb N \) and \(x \in \varDelta _0\). Using (iii) and applying a similar reasoning as above, we infer that there exists a set \(\varDelta _1 \in {\fancyscript{A}}\) such that \(\mu (X\setminus \varDelta _1)=0\) and \([\mathsf{h }_{\phi ^{i+j+1}}(x)]_{i,j=0}^n {\geqslant }0\) for all \(n \in \mathbb N \) and \(x \in \varDelta _1\). Employing (2.5) yields (a).

(b) By (a), there exists \(\varDelta \in {\fancyscript{A}}\) such that \(\mu (X \!\setminus \! \varDelta )=0, \mathsf{h }_{\phi ^0}(x) = 1\) and \( \left\{ \mathsf{h }_{\phi ^n}(x) \right\} _{n=0}^\infty \) is a Stieltjes moment sequence for every \(x\in \varDelta \). Hence, for every \(x\in \varDelta \) there exists a Borel probability measure \(\mu _x\) on \(\mathbb{R }_+\) such that \(\mathsf{h }_{\phi ^n}(x) = \int _{\mathbb{R }_+} s^n \mathrm{d}\mu _x(s)\) for all \(n \in \mathbb{Z }_+\). This yields

$$\begin{aligned} \bigg (\sum _{j=0}^n \mathsf{h }_{\phi ^j}\bigg )(x)&= \int _{\mathbb{R }_+} \bigg (\sum _{j=0}^n s^j\bigg ) \mathrm{d}\mu _x(s)\\&= \int _{[0,1)} \bigg (\sum _{j=0}^n s^j\bigg ) \mathrm{d}\mu _x(s) + \int _{[1,\infty )} \bigg (\sum _{j=0}^n s^j\bigg ) \mathrm{d}\mu _x(s)\\&{\leqslant }(n+1) \int _{[0,1)} 1 \mathrm{d}\mu _x(s) + (n+1) \int _{[1,\infty )} s^n \mathrm{d}\mu _x(s)\\&{\leqslant }(n+1) (1 + \mathsf{h }_{\phi ^n}) (x), \quad x \in \varDelta , \, n \in \mathbb N . \end{aligned}$$

Hence, the domains of \(C_{\phi }^n\) and \(C_{\phi ^n}\) coincide for all \(n\in \mathbb N \). By (3.3), this gives (b).

(c) Apply (i) and Theorem 4.7. This completes the proof. \(\square \)

Lemma 10.2

Suppose that \(\phi \) is a nonsingular transformation of \(X\) satisfying the following two conditions:

  1. (i)

    \(\mathcal{D }(C_\phi ^n)\) is dense in \(L^2(\mu )\) for every \(n\in \mathbb N \),

  2. (ii)

    \(\big [\Vert C_\phi ^{i+j}f\Vert ^2\big ]_{i,j=0}^n {\geqslant }0\) for all \(f\in \mathcal{D }(C_\phi ^{2n})\) and \(n\in \mathbb N \).

Then the assertions (a), (b) and (c) of Lemma 10.1 hold.

Proof

Set \(\mathcal{E }_n=\mathcal{D }(C_\phi ^n)\) for \(n\in \mathbb N \). According to (3.6), each \(\mathcal{E }_n\) satisfies (12.5). Substituting \(C_\phi f\) for \(f\) in (ii) implies that the hypothesis (iii) of Lemma 10.1 is satisfied. Applying Lemma 10.1 completes the proof. \(\square \)

Corollary 10.3

If \(C_\phi \) is subnormal and \(\overline{\mathcal{D }(C_\phi ^n)}=L^2(\mu )\) for all \(n \in \mathbb N \), then the assertions (a), (b) and (c) of Lemma 10.1 hold.

Proof

Apply Proposition 2.4 and Lemma 10.2. \(\square \)

The following theorem completely characterizes composition operators that generate Stieltjes moment sequences. It should be compared with Lambert’s characterizations of bounded subnormal composition operators (cf. [31]). In particular, condition (ii) of Theorem 10.4 is the Lambert condition, which in the bounded case is equivalent to subnormality.

Theorem 10.4

If \(\phi \) is a nonsingular transformation of \(X\), then the following conditions are equivalent:

  1. (i)

    \(C_\phi \) generates Stieltjes moment sequences,

  2. (ii)

    \( \left\{ \mathsf{h }_{\phi ^n}(x) \right\} _{n=0}^\infty \) is a Stieltjes moment sequence for \(\mu \)-a.e. \(x \in X\),

  3. (iii)

    \(\overline{\mathcal{D }(C_\phi ^k)} = L^2(\mu )\) for all \(k\in \mathbb N \), and \(\{\mu (\phi ^{-n}(\varDelta ))\}_{n=0}^\infty \) is a Stieltjes moment sequence for every \(\varDelta \in {\fancyscript{A}}\) such that \(\mu (\phi ^{-k}(\varDelta )) < \infty \) for all \(k \in \mathbb{Z }_+\),

  4. (iv)

    \(\mathsf{h }_{\phi ^n} < \infty \) a.e. \([\mu ]\) for all \(n\in \mathbb N \) and \(L(p){\geqslant }0\) a.e. \([\mu ]\) whenever \(p(t) {\geqslant }0\) for all \(t \in \mathbb{R }_+\), where \(L:\mathbb{C }[t] \rightarrow \mathcal{M }\) is a linear mapping determined byFootnote 6

    $$\begin{aligned} L(t^n) = \mathsf{h }_{\phi ^n}, \quad n \in \mathbb{Z }_+; \end{aligned}$$

    here \(\mathbb{C }[t]\) is the set of all complex polynomials in one real variable \(t\) and \(\mathcal{M }\) is the set of all \({\fancyscript{A}}\)-measurable complex functions on \(X\).

Moreover, if (i) holds, then \(C_\phi ^n=C_{\phi ^n}\) and \(\mathcal{D }^\infty (C_\phi )\) is a core for \(C_\phi ^n\) for all \(n\in \mathbb{Z }_+\).

Proof

(i)\(\Rightarrow \)(ii) Set \(\mathcal{E }_n=\mathcal{D }^\infty (C_\phi )\) for \(n \in \mathbb N \). By (2.5), (3.6) and Lemma 10.1, we see that the condition (ii) and the “moreover” part hold.

(ii)\(\Rightarrow \)(i) Take \(f \in \mathcal{D }^\infty (C_\phi ), n \in \mathbb{Z }_+\) and \(\{\alpha _i\}_{i=0}^n\subseteq \mathbb{C }\). Then, by (2.5), we have

$$\begin{aligned} \sum _{i,j=0}^n \alpha _i \overline{\alpha }_j \Vert C_{\phi }^{i+j} f\Vert ^2 \overset{(3.4)}{=} \int _X \bigg (\sum _{i,j=0}^n \alpha _i \overline{\alpha }_j \mathsf{h }_{\phi ^{i+j}}(x)\bigg ) |f(x)|^2 \mathrm{d}\mu (x) {\geqslant }0. \end{aligned}$$

Applying the above to \(C_\phi f\) in place of \(f\), we deduce that the sequences \(\{\Vert C_\phi ^k f\Vert ^2\}_{k=0}^\infty \) and \(\{\Vert C_\phi ^{k+1} f\Vert ^2\}_{k=0}^\infty \) are positive definite. Therefore, by (2.5), \(\{\Vert C_\phi ^k f\Vert ^2\}_{k=0}^\infty \) is a Stieltjes moment sequence. It follows from Corollary 4.5(ii) and Theorem 4.7 that \(\mathcal{D }^\infty (C_\phi )\) is dense in \(L^2(\mu )\).

(i)\(\Rightarrow \)(iii) Evident (because \(\chi _\varDelta \in \mathcal{D }^\infty (C_\phi )\) for every \(\varDelta \) as in (iii)).

(iii)\(\Rightarrow \)(i) By Theorem 4.7, the set \(\mathcal{D }^\infty (C_\phi )\) is dense in \(L^2(\mu )\). Consider a simple \({\fancyscript{A}}\)-measurable function \(u =\sum _{i=1}^k \alpha _i \chi _{\varDelta _i}\), where \(\{\alpha _i\}_{i=1}^k\) are positive real numbers and \(\{\varDelta _i\}_{i=1}^k\) are pairwise disjoint sets in \({\fancyscript{A}}\). Suppose that \(u\) is in \(\mathcal{D }^\infty (C_\phi )\). Then, by the measure transport theorem, \(\left\{ \chi _{\varDelta _i} \right\} _{i=1}^k \subseteq \mathcal{D }^\infty (C_\phi )\) and

$$\begin{aligned} \Vert C_\phi ^n u\Vert ^2 = \sum _{i,j=1}^k \alpha _i \alpha _j \int \limits _{\varDelta _i \cap \varDelta _j} \mathsf{h }_{\phi ^n} \mathrm{d}\mu = \sum _{i=1}^k \alpha _i^2 \int \limits _{\varDelta _i} \mathsf{h }_{\phi ^n} \mathrm{d}\mu \overset{(3.4)}{=} \sum _{i=1}^k \alpha _i^2 \mu (\phi ^{-n}(\varDelta _i)) \end{aligned}$$

for all \(n\in \mathbb{Z }_+\). Hence, by (iii), we have

$$\begin{aligned}&\{\Vert C_\phi ^n u\Vert ^2\}_{n=0}^\infty \, \text{ is} \text{ a} \text{ Stieltjes} \text{ moment} \text{ sequence} \text{ for} \text{ every} \text{ simple}\nonumber \\&\text{ nonnegative}\; {\fancyscript{A}}\text{-measurable} \text{ function}\, u\in \mathcal{D }^\infty (C_\phi ). \end{aligned}$$
(10.2)

Now take \(f \in \mathcal{D }^\infty (C_\phi )\). Then, there exists a sequence \(\{u_n\}_{n=1}^\infty \) of simple \({\fancyscript{A}}\)-measurable functions \(u_n:X \rightarrow \mathbb{R }_+\) such that \(u_n(x) {\leqslant }u_{n+1}(x) {\leqslant }|f(x)|\) and \(\lim _{k \rightarrow \infty } u_k(x) = |f(x)|\) for all \(n\in \mathbb N \) and \(x \in X\). This implies that \(\{u_n\}_{n=1}^\infty \subseteq \mathcal{D }^\infty (C_\phi )\) and, by Lebesgue’s monotone convergence theorem,

$$\begin{aligned} \Vert C_\phi ^n f\Vert ^2 = \int \limits _X |f|^2 \mathsf{h }_{\phi ^n} \mathrm{d}\mu = \lim _{k\rightarrow \infty } \int \limits _X u_k^2 \mathsf{h }_{\phi ^n} \mathrm{d}\mu = \lim _{k\rightarrow \infty } \Vert C_\phi ^n u_k\Vert ^2, \quad n \in \mathbb{Z }_+. \end{aligned}$$

Since the class of Stieltjes moment sequences is closed under the operation of taking pointwise limits (cf. (2.5)), we infer from (10.2) that \(\{\Vert C_\phi ^n f\Vert ^2\}_{n=0}^\infty \) is a Stieltjes moment sequence.

(ii)\(\Rightarrow \)(iv) If \(p\in \mathbb{C }[t]\) is such that \(p(t) {\geqslant }0\) for all \(t \in \mathbb{R }_+\), then there exist \(q_1,q_2\in \mathbb{C }[t]\) such that \(p(t)=t|q_1(t)|^2 + |q_2(t)|^2\) for all \(t \in \mathbb{R }\) (see [36, Problem 45, p. 78]). This fact combined with (2.5) implies that \(L(p) {\geqslant }0\) a.e. \([\mu ]\).

(iv)\(\Rightarrow \)(ii) Let \(Q\) be a countable dense subset of \(\mathbb{C }\). If \(q\in \mathbb{C }[t]\) is a polynomial with coefficients in \(Q\), then the polynomials \(p_1:=|q|^2\) and \(p_2:=t|q|^2\) are nonnegative on \(\mathbb{R }_+\). Hence, \(L(p_i) {\geqslant }0\) a.e. \([\mu ]\) for \(i=1,2\). Since \(Q\) is countable, this implies that there exists \(\varDelta \in {\fancyscript{A}}\) such that \(\mu (X\setminus \varDelta )=0\),

$$\begin{aligned} 0 \le \mathsf{h }_{\phi ^n}(x)<\infty , \sum _{i,j=0}^n \alpha _i \overline{\alpha }_j \mathsf{h }_{\phi ^{i+j}}(x) {\geqslant }0 \text{ and} \sum _{i,j=0}^n \alpha _i \overline{\alpha }_j \mathsf{h }_{\phi ^{i+j+1}}(x) {\geqslant }0 \end{aligned}$$
(10.3)

for all \(n \in \mathbb{Z }_+, \{\alpha _i\}_{i=0}^n \subseteq Q\) and \(x \in \varDelta \). As \(Q\) is dense in \(\mathbb{C }\), we see that (10.3) holds for all \(n \in \mathbb{Z }_+, \{\alpha _i\}_{i=0}^n \subseteq \mathbb{C }\) and \(x \in \varDelta \). This and (2.5) complete the proof. \(\square \)

11 Conclusion

We close the paper by pointing out that there exists a composition operator generating Stieltjes moment sequences which is not subnormal and even not hyponormal. Such an operator can be constructed on the basis of a weighted shift on a directed tree with one branching vertex (cf. [25, Section 4.3]). In view of Theorem 10.4, any composition operator \(C_\phi \) which generates Stieltjes moment sequences, in particular the aforementioned, satisfies the conditions (ii), (iii) and (iv) of this theorem as well as its “moreover” part (specifically, \(\mathcal{D }^\infty (C_\phi )\) is a core for \(C_\phi ^n\) for every \(n\in \mathbb{Z }_+\), which is considerably more than is required in Definition 2.3). Therefore, none of the Lambert characterizations of subnormality of bounded composition operators (cf. [31]) is valid in the unbounded case. It is worth mentioning that the above example is built over the discrete measure space. However, it can be immediately adapted to the context of measures which are equivalent to the Lebesgue measure on \([0,\infty )\) by applying [24, Theorem 2.7].