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

1. 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 [1]; see also [2] for a different construction). More than four decades later, Schmüdgen discovered another pathological behaviour of domains of powers of symmetric operators (cf. [3]). It is well known that symmetric operators are subnormal (cf. [4, Theorem 1 in Appendix I.2]). Hence, closed subnormal operators may have nondensely defined powers. In turn, quasinormal operators, which are subnormal as well (see [5, 6]), have all powers densely defined (cf. [6]). In the present paper we discuss the above question in the context of subnormal weighted shifts on directed trees and subnormal composition operators in L2-spaces (over σ-finite measure spaces).

As recently shown (cf. [7, Proposition 3.1]), formally normal (in particular symmetric) weighted shifts on directed trees are automatically bounded and normal (in general, formally normal operators are not subnormal, cf. [8]). The same applies to symmetric composition operators in L2-spaces (cf. [9, Proposition B.1]). Formally normal composition operators in L2-spaces, which may be unbounded (see [9, Appendix C]), are still normal (cf. [10, Theorem 9.4]). As a consequence, all powers of such operators are densely defined (see e.g., [11, Corollary 5.28]).

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 L2-spaces. To answer both of them, we proceed as follows. First, by applying a recently established criterion for subnormality of weighted composition operators in L2-spaces which makes no appeal to density of C∞-vectors (see Theorem 1), we show that a densely defined weighted shift on a directed tree which admits a consistent system of probability measures (i.e., a system {μv}v∈V of Borel probability measures on ℝ+ which satisfies (6)) 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). The particular case of directed trees with one branching vertex is examined in Theorem 5 and Corollary 6. Using these two results, we answer both questions in the affirmative (see Example 1 and Remark 8). 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., [12, 13]).

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 δt 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). Set 𝒟∞(A)=⋂n=0∞𝒟(An). Recall that a closed densely defined operator A in ℋ is said to be normal if AA*=A*A (see [11, 14, 15] for more on this class of operators). We say that a densely defined operator A in ℋ is subnormal if there exist a complex Hilbert space 𝒦 and a normal operator N in 𝒦 such that ℋ⊆𝒦 (isometric embedding) and Ah=Nh for all h∈𝒟(S). We refer the reader to [6, 16–19] for the foundations of the theory of bounded and unbounded subnormal operators, respectively.

2. Weighted Composition Operators

Assume that (X,𝒜,ν) is a σ-finite measure space, w:X→ℂ is an 𝒜-measurable function, and ϕ:X→X is an 𝒜-measurable mapping. Define the σ-finite measure νw:𝒜→ℝ¯+ by νw(Δ)=∫Δ|w|2dν for Δ∈𝒜. Let νw∘ϕ-1:𝒜→ℝ¯+ be the measure given by νw∘ϕ-1(Δ)=νw(ϕ-1(Δ)) for Δ∈𝒜. Assume that νw∘ϕ-1 is absolutely continuous with respect to ν. By the Radon-Nikodym theorem (cf. [20, Theorem 2.2.1]), there exists a unique (up to a.e. [ν] equivalence) 𝒜-measurable function h=hϕ,w:X→ℝ¯+ such that
(1)νw∘ϕ-1(Δ)=∫Δhdν,Δ∈𝒜.
Then the operator C=Cϕ,w in L2(ν), given by
(2)𝒟(C)={f∈L2(ν):w·(f∘ϕ)∈L2(ν)},Cf=w·(f∘ϕ),f∈𝒟(C),
is well defined (cf. [21, Proposition 7]). Call C a weighted composition operator. By [21, Proposition 10], C is densely defined if and only if h<∞ a.e. [ν]; moreover, if this is the case, then νw|ϕ-1(𝒜) is σ-finite and, by the Radon-Nikodym theorem, for every 𝒜-measurable function f:X→ℝ¯+ there exists a unique (up to a.e. [νw] equivalence) ϕ-1(𝒜)-measurable function E(f)=Eϕ,w(f):X→ℝ¯+ such that
(3)∫ϕ-1(Δ)fdνw=∫ϕ-1(Δ)E(f)dνw,Δ∈𝒜.
We call E(f) the conditional expectation of f with respect to ϕ-1(𝒜) (see [21] for more information). A mapping P:X×𝔅(ℝ+)→[0,1] is called an 𝒜-measurable family of probability measures if the set-function P(x,·) 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 [21, Theorem 29].

Theorem 1.

If C is densely defined, h>0 a.e. [νw], and there exists an 𝒜-measurable family of probability measures P:X×𝔅(ℝ+)→[0,1] such that
(CC)E(P(·,σ))(x)=∫σtP(ϕ(x),dt)h(ϕ(x)),hforνw-a.e.x∈X,σ∈𝔅(ℝ+),
then C is subnormal.

Regarding Theorem 1, recall that if C is subnormal, then h>0 a.e. [νw] (cf. [21, Corollary 13]).

3. 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, resp.). Set Chi(u)={v∈V:(u,v)∈E} for u∈V. Denote by par the partial function from V to V which assigns to a 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. Set V∘=V∖{root} if 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 [12] for all facts about directed trees needed in this paper.

By a weighted shift on 𝒯 with weights λ={λv}v∈V∘⊆ℂ we mean the operator Sλ in ℓ2(V) defined by
(4)𝒟(Sλ)={f∈ℓ2(V):Λ𝒯f∈ℓ2(V)},Sλf=Λ𝒯f,f∈𝒟(Sλ),
where Λ𝒯 is the mapping defined on functions f:V→ℂ via
(5)(Λ𝒯f)(v)={λv·f(par(v)),ifv∈V∘,0,ifv=root.
As usual, ℓ2(V) is the Hilbert space of square summable complex functions on V with standard inner product. For u∈V, we define eu∈ℓ2(V) to be the characteristic function of the one-point set {u}. Then {eu}u∈V is an orthonormal basis of ℓ2(V).

The following useful lemma is an extension of part (iv) of [13, Theorem 3.2.2].

Lemma 2.

Let Sλ be a weighted shift on a directed tree 𝒯=(V,E) with weights λ={λv}v∈V∘ and let n∈ℤ+. Then Sλn is densely defined if and only if eu∈𝒟(Sλn) for every u∈V≺.

Proof.

In view of [13, Theorem 3.2.2(iv)], Sλn is densely defined if and only if eu∈𝒟(Sλn) for every u∈V. Note that if u∈V and Chi(u)={v}, then eu∈𝒟(Sλ) and Sλeu=λvev, which implies that eu∈𝒟(Sλn+1) whenever ev∈𝒟(Sλn). In turn, if Chi(u)=⌀, then clearly eu∈𝒟∞(Sλ). Using the above and an induction argument (related to paths in 𝒯), we deduce that Sλn is densely defined if and only if eu∈𝒟(Sλn) for every u∈V≺.

It is worth mentioning that if V≺=⌀, then, by Lemma 2 and [13, Theorem 3.2.2(iv)] (or by the proof of Lemma 2), 𝒟∞(Sλ) is dense in ℓ2(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 [22, Theorem 5.1.1], we do not assume the density of C∞-vectors in the underlying ℓ2-space. Moreover, we do not assume that the underlying directed tree is rootless and leafless, which is required in [9, 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. [12, Proposition 3.1.10]). Here, and later, we adopt the conventions that 0·∞=∞·0=0, 1/0=∞ and ∑v∈ϕξv=0; we also write ∫0∞ in place of ∫ℝ+.

Theorem 3.

Let Sλ be a weighted shift on a countably infinite directed tree 𝒯=(V,E) with weights λ={λv}v∈V∘. Suppose that there exist a system {μv}v∈V of Borel probability measures on ℝ+ and a system {εv}v∈V of nonnegative real numbers such that
(6)μu(σ)=∑v∈Chi(u)|λv|2∫σ1tμv(dt)+εuδ0(σ),hhhhhhhhhhhhhihhhσ∈𝔅(ℝ+),u∈V.
Then the following two assertions hold:

if Sλ is densely defined, then Sλ is subnormal,

if n∈ℕ, then Sλn is densely defined if and only if ∫0∞sndμu(s)<∞ for all u∈V≺.

Proof.

(i) Assume that Sλ is densely defined. Set X=V and 𝒜=2V. 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
(7)w(x)={λx,ifx∈V∘,0,ifx=root,ϕ(x)={par(x),ifx∈V∘root,ifx=root.
Clearly, the measure νw∘ϕ-1 is absolutely continuous with respect to ν and
(8)h(x)=νw(ϕ-1({x}))=νw(Chi(x))=∑y∈Chi(x)|λy|2,hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhx∈X.
Thus, by [12, Proposition 3.1.3], h(x)<∞ for every x∈X. We claim that h>0 a.e. [νw]. This is the same as to show that if x∈V∘ and νw(Chi(x))=0, then λx=0. Thus, if x∈V∘ and νw(Chi(x))=0, then applying (6) to u=x, we deduce that μx=δ0; in turn, applying (6) to u=par(x) with σ={0}, we get λx=0, which proves our claim.

Note that X=⨆x∈Xϕ-1({x}) (the disjoint union). Hence, the conditional expectation E(f) of a function f:X→ℝ¯+ with respect to ϕ-1(𝒜) is given by
(9)E(f)(z)=∫Chi(x)fdνwh(x),z∈ϕ-1({x}),x∈X+,
where X+∶={x∈X:νw(Chi(x))>0} (see also (8)); on the remaining part of X we can put E(f)=0.

Substituting σ={0} into (6), we see that μy({0})=0 for every y∈V∘ such that λy≠0. Thus, using the standard measure-theoretic argument and (6), we deduce that
(10)∫σtdμx(t)=∑y∈Chi(x)|λy|2μy(σ),σ∈𝔅(ℝ+),x∈X.
Set P(x,σ)=μx(σ) for x∈X and σ∈𝔅(ℝ+). It follows from (9) and (10) that P : X×𝔅(ℝ+)→[0,1] is a (𝒜-measurable) family of probability measures which fulfils the following equality:
(11)E(P(·,σ))(z)=∫σtP(ϕ(z),dt)h(ϕ(z)),z∈ϕ-1({x}),x∈X+.
This implies that P satisfies (CC). Hence, by Theorem 1, the weighted composition operator C (see (2)) is subnormal. Since Sλ=C, assertion (i) is proved.

(ii) It is easily seen that if μ is a finite positive Borel measure on ℝ+ and ∫0∞sndμ(s)<∞ for some n∈ℕ, then ∫0∞skdμ(s)<∞ for every k∈ℕ such that k⩽n. This fact combined with Lemma 2 and [22, Lemmata 2.3.1(i) and 4.2.2(i)] implies assertion (ii).

Remark 4.

Assume that Sλ is a densely defined weighted shift on a countably infinite directed tree 𝒯=(V,E) with weights λ={λv}v∈V∘. A careful inspection of the proof of Theorem 3 reveals that if {μx}x∈X (with X=V) is a system of Borel probability measures on ℝ+ which satisfies (6), then h>0 a.e. [νw], the family P defined by P(x,·)=μx for x∈X satisfies (CC), and μx=δ0 for every x∈X∖X+. We claim that if h>0 a.e. [νw] 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 by
(12)μ~x={P(x,·),ifx∈X+,δ0,otherwise,
satisfies (6) with {μ~x}x∈X in place of {μx}x∈X. Indeed, (CC) implies (11). Hence, by (9), equality in (10) holds for every x∈X+ with μz=P(z,·) for z∈X. This implies via the standard measure-theoretic argument that equality in (6) holds for every u∈X+. Since h>0 a.e. [νw], we deduce that equality in (6) holds for every u∈X+ with {μ~x}x∈X in place of {μx}x∈X. Clearly, this is also the case for u∈X∖X+. Thus, our claim is proved.

4. Trees with One Branching Vertex

Theorem 3 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). For η,κ∈ℤ+⊔{∞} with η⩾2, we define the directed tree 𝒯η,κ=(Vη,κ,Eη,κ) as follows (the symbol “⊔” denotes disjoint union of sets):
(13)Vη,κ={-k:k∈Jκ}⊔{0}⊔{(i,j):i∈Jη,j∈ℕ},Eη,κ=Eκ⊔{(0,(i,1)):i∈Jη}00000⊔{((i,j),(i,j+1)):i∈Jη,j∈ℕ},Eκ={(-k,-k+1):k∈Jκ},
where Jn={k∈ℕ:k⩽n} for n∈ℤ+⊔{∞}. Clearly, 𝒯η,κ is leafless and 0 is its only branching vertex. From now on, we write λi,j instead of the more formal expression λ(i,j) whenever (i,j)∈Vη,κ.

Theorem 5.

Let η,κ∈ℤ+⊔{∞} be such that η⩾2 and let Sλ be a weighted shift on a directed tree 𝒯η,κ with nonzero weights λ={λv}v∈Vη,κ∘. Suppose that there exists a sequence {μi}i=1η of Borel probability measures on ℝ+ such that
(14)∫0∞sndμi(s)=|∏j=2n+1λi,j|2,n∈ℕ,i∈Jη,
and that one of the following three disjunctive conditions is satisfied:

κ=0 and
(15)∑i=1η|λi,1|2∫0∞1sdμi(s)⩽1,

0<κ<∞ and
(16)∑i=1η|λi,1|2∫0∞1sdμi(s)=1,(17)|∏j=0l-1λ-j|2∑i=1η|λi,1|2∫0∞1sl+1dμi(s)=1,l∈Jκ-1,(18)|∏j=0κ-1λ-j|2∑i=1η|λi,1|2∫0∞1sκ+1dμi(s)⩽1,

κ=∞ and equalities (16) and (17) are valid.

Then the following two assertions hold:

if Sλ is densely defined, then Sλ is subnormal,

if n∈ℕ, then Sλn is densely defined if and only if
(19)∑i=1η|λi,1|2∫0∞sn-1dμi(s)<∞.

Proof.

As in the proof of [23, Theorem 4.1], we define the system {μv}v∈Vη,κ of Borel probability measures on ℝ+ and verify that {μv}v∈Vη,κ satisfies (6). Hence, assertion (a) is a direct consequence of Theorem 3(i).

(b) Fix n∈ℕ. It follows from Theorem 3(ii) that Sλn is densely defined if and only if ∫0∞sndμ0(s)<∞. Using the explicit definition of μ0 and applying the standard measure-theoretic argument, we see that
(20)∫0∞sndμ0(s)=∑i=1η|λi,1|2∫0∞sn-1dμi(s).
This completes the proof of assertion (b) (the case of n=1 can also be settled without using the definition of μ0 simply by applying Lemma 2 and [12, Proposition 3.1.3(iii)]).

Note that Theorem 5 remains true if its condition (ii) is replaced by the condition (iii) of [23, Theorem 4.1] (see also [23, Lemma 4.2] and its proof).

Corollary 6.

Under the assumptions of Theorem 5, if n∈ℕ, then the following two assertions are equivalent:

Sλn is densely defined and Sλn+1 is not,

the condition (19) holds and ∑i=1η|λi,1|2∫0∞sndμi(s)=∞.

5. The Example

It follows from [22, Lemma 2.3.1(i)] that if Sλ is a weighted shift on 𝒯η,κ and η<∞, then 𝒟∞(Sλ) is dense in ℓ2(Vη,κ) (this means that Corollary 6 is interesting only if η=∞). If η=∞, the situation is completely different. Using Theorem 5 and Corollary 6, we show that for every n∈ℕ and for every κ∈ℤ+⊔{∞}, there exists a subnormal weighted shift Sλ on 𝒯∞,κ such that Sλn is densely defined and Sλn+1 is not. For this purpose, we adapt [12, Procedure 6.3.1] to the present context. In the original procedure, one starts with a sequence {μi}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={λv}v∈V∞,κ∘ that satisfies the assumptions of Theorem 5 (in fact, using Lemma 7 below, we can also maintain the condition (19)). However, in general, it is not possible to maintain the condition (ii) of Corollary 6 even if {μi}i=1∞ are measures with two-point supports (this question is not discussed here).

Example 1.

Assume that η=∞. Consider the measures μi=δqi with qi∈(0,∞) for i∈ℕ. By [12, Notation 6.1.9 and Procedure 6.3.1], Sλ∈B(ℓ2(V∞,κ)) if and only if sup{qi:i∈ℕ}<∞. Hence, there is no loss of generality in assuming that sup{qi:i∈ℕ}=∞. To cover all possible choices of κ∈ℤ+⊔{∞}, we look for a system of nonzero weights {λv}v∈V∞,∞ which satisfies (14), (16), (17) with κ=∞, (19) and the equality ∑i=1∞|λi,1|2∫0∞sndμi(s)=∞. Setting λi,1=αiqi for i∈ℕ, we reduce our problem to find a sequence {αi}i=1∞⊆(0,∞) such that
(21)∑i=1∞αiqil<∞,l∈ℤ,l⩽n,∑i=1∞αiqin+1=∞.
Indeed, if {αi}i=1∞ is such a sequence, then multiplying its terms by an appropriate positive constant, we may assume that {αi}i=1∞ satisfies (21) and (16). Next we define the weights {λ-j:j∈ℤ+} recursively so as to satisfy (17) with κ=∞, and finally we set λi,j=qi for all i,j∈ℕ such that j⩾2. The so constructed weights {λv}v∈V∞,∞ meet our requirements.

The following lemma turns out to be helpful when solving the reduced problem.

Lemma 7.

If [ai,j]i,j=1∞ is an infinite matrix with entries ai,j∈ℝ+, then there exists a sequence {αi}i=1∞⊆(0,∞) such that
(22)∑i=1∞αiai,j<∞,j∈ℕ.

Proof.

First observe that, for every i∈ℕ, there exists αi∈(0,∞) such that αi∑k=1iai,k⩽2-i. Hence, ∑i=j∞αiai,j⩽1 for every j∈ℕ.

Since sup{qi:i∈ℕ}=∞, there exists a subsequence {qik}k=1∞ of the sequence {qi}i=1∞ such that qik⩾k for every k∈ℕ. Set Ω={ik:k∈ℕ}. By Lemma 7, there exists {αi}i∈ℕ∖Ω⊆(0,∞) such that
(23)∑i∈ℕ∖Ωαiqil<∞,l∈ℤ,l⩽n.
Define the system {αi}i∈Ω⊆(0,∞) by
(24)αik=1k2qikn,k∈ℕ.
Since qik⩾k for all k∈ℕ, we get
(25)∑i∈Ωαiqil=∑k=1∞αikqikl=∑k=1∞1k2qikn-l⩽∑k=1∞1k2<∞,l∈ℤ,l⩽n,∑i∈Ωαiqin+1=∑k=1∞αikqikn+1=∑k=1∞qikk2⩾∑k=1∞1k=∞.
Combining (23) and (25), we get (21), which solves the reduced problem and consequently gives the required example.

Remark 8.

It is worth mentioning that if κ=∞, then any weighted shift Sλ on 𝒯∞,∞ with nonzero weights is unitarily equivalent to an injective composition operator in an L2-space over a σ-finite measure space (cf. [13, Lemma 4.3.1]). This fact combined with Example 1 shows that, for every n∈ℕ, there exists a subnormal composition operator C in an L2-space over a σ-finite measure space such that Cn is densely defined and Cn+1 is not.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The research of the first author was supported by the NCN (National Science Center) Grant DEC-2011/01/D/ST1/05805. The research of the third and the fourth authors was supported by the MNiSzW (Ministry of Science and Higher Education) Grant NN201 546438 (2010–2013).

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