Subnormal weighted shifts on directed trees and composition operators in $L^2$ spaces with non-densely defined powers

It is shown that for every positive integer $n$ there exists a subnormal weighted shift on a directed tree (with or without root) whose $n$th power is densely defined while its $(n+1)$th power is not. As a consequence, for every positive integer $n$ there exists a non-symmetric subnormal composition operator $C$ in an $L^2$ space over a $\sigma$-finite measure space such that $C^n$ is densely defined and $C^{n+1}$ is not.


Introduction
The question of when powers of a closed densely defined linear operator are densely defined has attracted considerable attention. In 1940 Naimark gave a surprising example of a closed symmetric operator whose square has trivial domain (see [16]; see also [10] for a different construction). More than four decades later, Schmüdgen discovered another pathological behaviour of domains of powers of symmetric operators (cf. [17]). It is well-known that symmetric operators are subnormal (cf. [1, Theorem 1 in Appendix I.2]). Hence, closed subnormal operators may have non-densely defined powers. In turn, quasinormal operators, which are subnormal as well (see [4] and [20]), have all powers densely defined (cf. [20]). In the present paper we discuss the above question in the context of subnormal weighted shifts on directed trees and subnormal composition operators in L 2 spaces (over σ-finite measure spaces).
The above discussion suggests the question of whether for every positive integer n there exists a subnormal weighted shift on a directed tree whose nth power is densely defined while its (n + 1)th power is not. A similar question can be asked for composition operators in L 2 spaces. To answer both of them, we proceed as follows. First, by applying a recently established criterion for subnormality of weighted composition operators in L 2 spaces which makes no appeal to density of C ∞ -vectors (see Theorem 2.1), we show that a densely defined weighted shift on a directed tree which admits a consistent system of probability measures 1 is subnormal, and, what is more, its nth power is densely defined if and only if all moments of these measures up to degree n are finite (cf. Theorem 3.2). The particular case of directed trees with one branching vertex is examined in Theorem 4.1 and Corollary 4.2. Using these two results, we answer both questions in the affirmative (see Example 5.1 and Remark 5.3). It is worth pointing out that though directed trees with one branching vertex have simple structure, they provide many examples which are important in operator theory (see e.g., [13,14]). Now we introduce some notation and terminology. In what follows, Z, Z + , N, R + and C stand for the sets of integers, nonnegative integers, positive integers, nonnegative real numbers and complex numbers, respectively. Set R + = R + ∪{∞}. We write B(R + ) for the σ-algebra of all Borel subsets of R + . Given t ∈ R + , we denote by δ t the Borel probability measure on R + concentrated on {t}.
The domain of an operator A in a complex Hilbert space H is denoted by D(A) (all operators considered in this paper are linear).
Recall that a closed densely defined operator A in H is said to be normal if AA * = A * A (see [3,18,23] for more on this class of operators). We say that a densely defined operator A in H is subnormal if there exists a complex Hilbert space K and a normal operator N in K such that H ⊆ K (isometric embedding) and Ah = N h for all h ∈ D(S). We refer the reader to [12] and [19,20,21,22] for the foundations of the theory of bounded and unbounded subnormal operators, respectively.

Weighted composition operators
Assume that (X, A , ν) is a σ-finite measure space, w : X → C is an Ameasurable function and φ : X → X is an A -measurable mapping. Define the σfinite measure ν w : A → R + by ν w (∆) = ∆ |w| 2 d ν for ∆ ∈ A . Let ν w •φ −1 : A → R + be the measure given by ν w • φ −1 (∆) = ν w (φ −1 (∆)) for ∆ ∈ A . Assume that ν w • φ −1 is absolutely continuous with respect to ν. By the Radon-Nikodym theorem (cf. [ Then the operator C = C φ,w in L 2 (ν) given by We call E(f ) the conditional expectation of f with respect to φ −1 (A ) (see [9] for more information). A mapping P : is a probability measure for every x ∈ X and the function P (·, σ) is A -measurable for every σ ∈ B(R + ).
The following criterion (read: a sufficient condition) for subnormality of unbounded weighted composition operators is extracted from [9, Theorem 27].
and there exists an A -measurable family of probability measures P : then C is subnormal.

Weighted shifts on directed trees
Let T = (V, E) be a directed tree (V and E stand for the sets of vertices and edges of T , respectively). Set Chi(u) = {v ∈ V : (u, v) ∈ E} for u ∈ V . Denote by par the partial function from V to V which assigns to each vertex u ∈ V its parent par(u) (i.e. a unique v ∈ V such that (v, u) ∈ E). A vertex u ∈ V is called a root of T if u has no parent. A root is unique (provided it exists); we denote it by root.
T has a root and V • = V otherwise. We say that u ∈ V is a branching vertex of V , and write u ∈ V ≺ , if Chi(u) consists of at least two vertices. We refer the reader to [13] for all facts about directed trees needed in this paper.
By a weighted shift on T with weights (As usual, ℓ 2 (V ) is the Hilbert space of square summable complex functions on V with standard inner product.) For u ∈ V , we define e u ∈ ℓ 2 (V ) to be the characteristic function of the one-point set {u}. Then {e u } u∈V is an orthonormal basis of ℓ 2 (V ). The following useful lemma is an extension of part (iv) of [14, Theorem 3.2.2].
Lemma 3.1. Let S λ be a weighted shift on a directed tree T = (V, E) with weights λ = {λ v } v∈V • and let n ∈ Z + . Then S n λ is densely defined if and only if e u ∈ D(S n λ ) for every u ∈ V ≺ .
Proof. In view of [14, Theorem 3.2.2(iv)], S n λ is densely defined if and only if e u ∈ D(S n λ ) for every u ∈ V . Note that if u ∈ V and Chi(u) = {v}, then e u ∈ D(S λ ) and S λ e u = λ v e v , which implies that e u ∈ D(S n+1 λ ) whenever e v ∈ D(S n λ ). In turn, if Chi(u) = ∅, then clearly e u ∈ D ∞ (S λ ). Using the above and an induction argument (related to paths in T ), we deduce that S n λ is densely defined if and only if e u ∈ D(S n λ ) for every u ∈ V ≺ .
It is worth mentioning that if V ≺ = ∅, then, by Lemma 3.1 and [14, Theorem 3.2.2(iv)] (or by the proof of Lemma 3.1), D ∞ (S λ ) is dense in ℓ 2 (V ). In particular, this covers the case of classical weighted shifts and their adjoints. Now we give a criterion for subnormality of weighted shifts on directed trees. As opposed to [5, Theorem 5.1.1], we do not assume the density of C ∞ -vectors in the underlying ℓ 2 -space. Moreover, we do not assume that the underlying directed tree is rootless and leafless, which is required in [8,Theorem 47], and that weights are nonzero. The only restriction we impose is that the directed tree is countably infinite. This is always satisfied if the weighted shift in question is densely defined and has nonzero weights (cf. [13, Proposition 3.1.10]). (3.1) Then the following two assertions hold : Proof. (i) Assume that S λ is densely defined. Set X = V and A = 2 V . Let ν : A → R + be the counting measure on X (ν is σ-finite because V is countable). Define the weight function w : X → C and the mapping φ : X → X by Clearly, the measure ν w • φ −1 is absolutely continuous with respect to ν and Thus, by [13, Proposition 3.1.3], h(x) < ∞ for every x ∈ X. We claim that h > 0 a.e. [ν w ]. This is the same as to show that if x ∈ V • and ν w (Chi(x)) = 0, then λ x = 0. Thus, if x ∈ V • and ν w (Chi(x)) = 0, then applying (3.1) to u = x, we deduce that µ x = δ 0 ; in turn, applying (3.1) to u = par(x) with σ = {0}, we get λ x = 0, which proves our claim. 2 We adopt the conventions that 0 · ∞ = ∞ · 0 = 0, 1 0 = ∞ and v∈∅ ξv = 0. 3 Here, and later, ∞ 0 means integration over the set R + .
(ii) It is easily seen that if µ is a finite positive Borel measure on R + and  [ν w ], the family P defined by P (x, ·) = µ x for x ∈ X satisfies (CC) and µ x = δ 0 for every x ∈ X \ X + . We claim that if h > 0 a.e. [ν w ] and P : X × B(R + ) → [0, 1] is any family of probability measures which satisfies (CC), then the system {μ x } x∈X of probability measures defined bỹ (3.5). Hence, by (3.3), equality in (3.4) holds for every x ∈ X + with µ z = P (z, ·) for z ∈ X. This implies via the standard measure-theoretic argument that equality in (3.1) holds for every u ∈ X + . Since h > 0 a.e. [ν w ], we deduce that equality in (3.1) holds for every u ∈ X + with {μ x } x∈X in place of {µ x } x∈X . Clearly, this is also the case for u ∈ X \ X + . Thus, our claim is proved.

Trees with one branching vertex
Theorem 3.2 will be applied in the case of weighted shifts on leafless directed trees with one branching vertex. First, we recall the models of such trees (see Figure 1 below). For η, κ ∈ Z + ⊔ {∞} with η 2, we define the directed tree T η,κ = (V η,κ , E η,κ ) as follows (the symbol " ⊔ " denotes disjoint union of sets) where J n = {k ∈ N : k n} for n ∈ Z + ⊔ {∞}. Clearly, T η,κ is leafless and 0 is its only branching vertex. From now on, we write λ i,j instead of the more formal expression λ (i,j) whenever (i, j) ∈ V η,κ .  and that one of the following three disjunctive conditions is satisfied : (iii) κ = ∞ and equalities (4.2) and (4.3) are valid. Then the following two assertions hold : (a) if S λ is densely defined, then S λ is subnormal, Proof. As in the proof of [6, Theorem 4.1], we define the system {µ v } v∈Vη,κ of Borel probability measures on R + and verify that {µ v } v∈Vη,κ satisfies (3.1). Hence, assertion (a) is a direct consequence of Theorem 3.2(i).
(b) Fix n ∈ N. It follows from Theorem 3.2(ii) that S n λ is densely defined if and only if ∞ 0 s n d µ 0 (s) < ∞. Using the explicit definition of µ 0 and applying the standard measure-theoretic argument, we see that This completes the proof of assertion (b) (the case of n = 1 can also be settled without using the definition of µ 0 simply by applying

The example
It follows from [5, Lemma 2.3.1(i)] that if S λ is a weighted shift on T η,κ and η < ∞, then D ∞ (S λ ) is dense in ℓ 2 (V η,κ ) (this means that Corollary 4.2 is interesting only if η = ∞). If η = ∞, the situation is completely different. Using Theorem 4.1 and Corollary 4.2, we show that for every n ∈ N and for every κ ∈ Z + ⊔ {∞}, there exists a subnormal weighted shift S λ on T ∞,κ such that S n λ is densely defined and S n+1 λ is not. For this purpose, we adapt [13, Procedure 6.3.1] to the present context. In the original procedure, one starts with a sequence {µ i } ∞ i=1 of Borel probability measures on R + (whose nth moments are finite for every n ∈ Z such that n −(κ+1)) and then constructs a system of nonzero weights λ = {λ v } v∈V • ∞,κ that satisfies the assumptions of Theorem 4.1 (in fact, using Lemma 5.2 below, we can also maintain the condition (4.5)). However, in general, it is not possible to maintain the condition (ii) of Corollary 4.2 even if {µ i } ∞ i=1 are measures with twopoint supports (this question is not discussed here).
Example 5.1. Assume that η = ∞. Consider the measures µ i = δ qi with q i ∈ (0, ∞) for i ∈ N. By [13, Notation 6.1.9 and Procedure 6.3.1], S λ ∈ B(ℓ 2 (V ∞,κ )) if and only if sup q i : i ∈ N < ∞. Hence, there is no loss of generality in assuming that sup q i : i ∈ N = ∞. To cover all possible choices of κ ∈ Z + ⊔ {∞}, we look for a system of nonzero weights {λ v } v∈V∞,∞ which satisfies (4.1), (4.2), (4.3) with κ = ∞, (4.5) and the equality α i q l i < ∞, l ∈ Z and l n, is such a sequence, then multiplying its terms by an appropriate positive constant, we may assume that {α i } ∞ i=1 satisfies (5.1), (5.2) and (4.2). Next we define the weights λ −j : j ∈ Z + recursively so as to satisfy (4.3) with κ = ∞, and finally we set λ i,j = √ q i for all i, j ∈ N such that j 2. The so constructed weights {λ v } v∈V∞,∞ meets our requirements.
The following lemma turns out to be helpful when solving the reduced problem.
Proof. First observe that for every i ∈ N, there exists α i ∈ (0, ∞) such that Set Ω = {i k : k ∈ N}. By Lemma 5.2, there exists {α i } i∈N\Ω ⊆ (0, ∞) such that i∈N\Ω α i q l i < ∞, l ∈ Z and l n. Remark 5.3. It is worth mentioning that if κ = ∞, then any weighted shift S λ on T ∞,∞ with nonzero weights is unitarily equivalent to an injective composition operator in an L 2 space over a σ-finite measure space (cf. [14,Lemma 4.3.1]). This fact combined with Example 5.1 shows that for every n ∈ N, there exists a subnormal composition operator C in an L 2 space over a σ-finite measure space such that C n is densely defined and C n+1 is not.