# A Note on Unbounded Hyponormal Composition Operators inL2-Spaces

Authors: Piotr Budzyński
Journal: Journal of Function Spaces and Applications
 (2012)

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## Abstract

We construct an unbounded hyponormal composition operatorCϕ in L2-space such that the domains of Cϕ2 and Cϕ2 are trivial.


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## Body

## 1. Introduction

The class of unbounded composition operators inL2-spaces seems to be an interesting object of research. It contains some classical operators (e.g., the shift operator on weighted ℓ2 space), and, what is more important, it also contains operators with properties not known before, sometimes very surprising (e.g., a nonsubnormal operator which generates Stieltjes moment sequences, see [1]). The theory of it is in early stage of development, and the literature is minimal (cf. [2–5]).In the present paper we provide an example of a hyponormal composition operator acting inℓ2(μ) whose square has trivial domain. Existence of such example is relevant to the problem of characterizing subnormality of unbounded composition operators in L2-spaces. In a recent paper [4], it was shown that the celebrated Lambert's characterizations of subnormality for bounded composition operators (cf. [6]) are no longer valid in the unbounded case (see also [1]). The authors also elaborated on the role of density of ⋂n=1∞D(Cϕn) in the underlying L2-space. It is important because of all known general criteria for subnormality of unbounded operators which do not require the density of ⋂n=1∞D(Cϕn); some auxiliary objects are assumed to exist (cf. [7–10]). However, verifying these additional assumptions seems to be difficult in the context of composition operators. In view of this, the question of existence of a subnormal composition operator Cϕ such that ⋂n=1∞D(Cϕn) is not dense in the underlying L2-space arises naturally and appears to be very intriguing (no such example is known). It is well known that subnormal operators are hyponormal. Therefore, our example sheds some light on the problem.The first example of a nonsymmetric hyponormal operator such that the domain of its square is trivial was provided within the class of weighted shifts on directed trees (cf. [11]). As it turned out very recently (cf. [1, Lemma 4.3.1]), some weighted shifts on directed trees are unitarily equivalent to composition operators in ℓ2-spaces. Our construction is direct and independent of any connection between both the classes. It should be also mentioned that our example, which is build over a discrete measure space, can be adapted so as to obtain an example over a nonatomic measure space.


## 2. Preliminaries

We denote byℤ and ℕ the sets of integers and positive integers, respectfully. By ℤ+ we understand the set ℕ∪{0}. If X is a set, then χX stands for the characteristic function of X; the power set of X will be denoted by 2X.Ifℋ is a (complex) Hilbert space and A is a (linear) operator acting in ℋ, then D(A) stands for the domain of A and A* stands for the adjoint operator of A (if it exists). A densely defined operator A in ℋ is said to be hyponormal if D(A)⊆D(A*) and ∥A*f∥⩽∥Af∥ for all f∈D(A). Interested reader is referred to [12–14] for some pieces of the theory of unbounded hyponormal operators.We recall now some information on composition operators inL2-spaces. We restrict ourself to the case of composition operator induced by a transformation of a countable set because our example is built in this way (general setting is presented in [4, 15, 16]). Let X be a countably infinite set, let μ be a measure on 2X such that μ({x})>0 for every x∈X, and let ϕ:X→X. By ℓ2(μ) we denote the Hilbert space of all complex valued functions on X which are square-summable with respect to the measure μ. The following formula
(2.1)D(Cϕ)={f∈ℓ2(μ):f∘ϕ∈ℓ2(μ)},Cϕf=f∘ϕ,f∈D(Cϕ),
defines the so-called composition operator(2.2)Cϕ:ℓ2(μ)⊇D(Cϕ)⟶ℓ2(μ).
It is well known that Cϕ is a closed linear operator in ℓ2(μ). Typically, properties of Cϕ are written in terms of the canonical Radon-Nikodym derivative hϕ attached to Cϕ (cf. [2–4, 6, 17–22]). In our setting, hϕ is given by
(2.3)hϕ(x)=μ(ϕ-1({x}))μ({x}),x∈X.
The operator of conditional expectation is also frequently employed when investigating composition operators. Let us recall that for a σ-algebra 𝒜⊆2X generated by a partition {σι:ι∈ℐ} of X and for a nonnegative function f on X, the conditional expectation of f with respect to 𝒜 can be defined as follows:
(2.4)E(f∣𝒜)=∑ι∈ℐ‍1μ(σι)(∫σι‍fdμ)·χσι.
We refer the reader to [23] for background on conditional expectation.The following lemma delivers few basic properties of composition operators inℓ2(μ) which we use to obtain the example. The proofs are straightforward; nonetheless we include them for completeness. One more piece of notation will be useful: ℰ stands for the set {χ{x}:x∈X}.Lemma 2.1.
The following conditions are equivalent:(i)
Cϕ is densely defined.(ii)
ℰ⊆D(Cϕ).(iii)
μ(ϕ-1({x}))<∞ for all x∈X.
Moreover, ifCϕ is densely defined, then the following assertion holds:  (iv)
Cϕ is hyponormal if and only if
(2.5)1μ({x})∑y∈ϕ-1({x})‍μ({y})2μ(ϕ-1({y}))≤1,x∈ϕ(X).Proof.
By definition,ℰ⊆D(Cϕ) if and only if χ{x}∘ϕ∈ℓ2(μ) for all x∈X. Since
(2.6)∫‍|χ{x}∘ϕ|2dμ=∫‍|χϕ-1({x})|2dμ=μ(ϕ-1({x})),x∈X,
we see that conditions (ii) and (iii) are equivalent.
That (ii) implies (i) is obvious. The converse can be proved as follows. Letf∈D(Cϕ). Then f=∑x∈Xf‍f(x)·χ{x}, where Xf:={x∈X:f(x)≠0}. If x0∈Xf, then χ{x0}∈D(Cϕ) because of the following
(2.7)|f(x0)|2∫‍χ{x0}∘ϕdμ=∫‍|f∘ϕ|2·χϕ-1({x0})dμ≤∫‍|f∘ϕ|2dμ<∞.
If D(Cϕ)¯=ℓ2(μ), then clearly X=⋃f∈D(Cϕ)‍Xf. Therefore, χ{x}∈D(Cϕ) for every x∈X, which gives (ii).
For the proof of the “moreover” statement we recall a general characterization of hyponormality of a densely defined composition operator induced by a nonsingular transformation of a measure space(X,Σ,μ) (see [2, Corollary 6.7]): Cϕ is hyponormal if and only if
(2.8)hϕ∘ϕ·E(1hϕ)≤1;
the conditional expectation above is calculated with respect to the completion of the σ-subalgebra ϕ-1(Σ) of Σ. In our situation the σ-subalgebra is generated by the partition {ϕ-1({x}):x∈ϕ(X)}. The characterization above combined with (2.3) and (2.4) yields (iv).By inspecting the part “(i) implies (ii)” of the proof above we can get the following description of when the domain of a composition operator inℓ2(μ) is trivial.Corollary 2.2.
D
(
C
ϕ
)
=
{
0
} if and only if ℰ∩D(Cϕ)=∅.


## 3. The Example

The main result of this paper is the existence of a hyponormal composition operator whose square has trivial domain.Theorem 3.1.
There exists a countably infinite setX, a positive measure μ on 2X and a map ϕ:X→X such that the composition operator Cϕ acting in ℓ2(μ) is hyponormal and D(Cϕ2)=D(Cϕ2)={0}.Proof.
SetX=ℕ×{1,2}. Let {X(n,k):n∈ℕ,k=1,2} be a family of subsets of X defined according to the following scheme:
(3.1)X(1,k)={(3,k),(7,k),(12,k),…},X(2,k)={(1,k),(4,k),(8,k),(13,k),…},X(3,k)={(2,k),(5,k),(9,k),(14,k),…},X(4,k)={(6,k),(10,k),(15,k),(21,k),…},X(5,k)={(11,k),(16,k),(22,k),…}.X(5,k)2⋮
Then the sets X(n,k) are pairwise disjoint, infinite, and X=⋃n∈ℕ,k=1,2‍X(n,k).
Now, we define an atomic measureμ on X. For this we take any sequence {a(n)}n=1∞ of positive real numbers such that
(3.2)211≤2k+2(a(k))2≤a(k+1),k∈ℕ.
Clearly, {a(n)}n=1∞ is monotonically increasing and a(k)⩾2k+3 for all k∈ℕ. Suppose x∈X(n,1) for some n∈ℕ. Since X(n,1)={(k0,1),(k1,1),(k2,1),…}, where k0<k1<k2<⋯ are nonnegative integers, there exists i∈ℤ+ such that x=(ki,1). Then we put
(3.3)μ({x})=a(n)2i.
On the other hand, if x∈ℕ×{2}, then we set
(3.4)μ({x})=1.
Since X=⋃n∈ℕ,k=1,2‍X(n,k), the measure μ is defined on the whole of 2X. Observe that μ(X(n,1))=2a(n) for every n∈ℕ. In order to define a transformation ϕ of X, we note that X={xn:n∈ℕ}, where
(3.5)xn:={(1,2),ifn=1,(l+1,2),ifn=2l,l∈ℕ,n∈ℕ,(l,1),ifn=2l+1,l∈ℕ.
It is easy to see that xm∉X(m,1) for m∈ℕ. Moreover, if xi∈X(m,1) for some i,m∈ℕ, then i>m. Now, if x∈X(n,1) for some n∈ℕ, then we set
(3.6)ϕ(x)=xn,
if x=(m,2) for some m∈ℕ, then we put
(3.7)ϕ(x)=(m,1).
Again, equality X=⋃n∈ℕ,k=1,2‍X(n,k) yields that the transformation ϕ is defined on the whole of X.
Now, we derive some complementary information onϕ and μ. Choose x∈X. Then x=(p,k)∈X(n,k) for some p,n∈ℕ and k∈{1,2}. Simultaneously, x=xi for some i∈ℕ. Therefore, we gather that
(3.8)ϕ-1({x})={X(i,1),fork=2,X(i,1)∪{(p,2)},fork=1.
Furthermore, if y∈ϕ-1({x}), then either y=(p,2) or y=(r,1) for some r∈ℕ. In both cases y=xm for some m∈ℕ. This yields
(3.9)ϕ-1({y})={X(m,1)∪{(r,2)},fork=2,X(m,1),fork=1,y=(p,2),X(m,1)∪{(r,2)},fork=1,y≠(p,2).
Suppose now that x∈X(n,1). Then, as above, x=(p,1)=xi for p,i∈ℕ. Note that
(3.10)μ({x})≥a(n)2i≥12i.
Also, if y=xl∈ϕ-1({x})∖{(p,2)}, then l>i. This, (3.8), (3.9), definition of μ, and (3.2), implies that
(3.11)∑y∈ϕ-1({x})‍μ({y})2μ(ϕ-1({y}))=μ({(p,2)})2μ(ϕ-1({(p,2)}))+∑y∈X(i,1)‍μ({y})2μ(ϕ-1({y}))⩽1μ(X(m′,1))+∑y∈X(i,1)‍μ({y})2μ(X(i+1,1))⩽12m′+4+μ(X(i,1))2μ(X(i+1,1))≤12m′+4+12i+1,
where m′∈ℕ obeys (p,2)=xm′. For such an m′ we either have i=m′+2 (for p=1) or i=m′+3 (for p>1), thus we obtain
(3.12)∑y∈ϕ-1({x})‍μ({y})2μ(ϕ-1({y}))≤12i.
This together with (3.10) implies that for all x∈⋃n=1∞‍X(n,1) we have
(3.13)∑y∈ϕ-1({x})‍μ({y})2μ(ϕ-1({y}))≤μ({x}).
On the other hand, if x=xi∈X(n,2), then μ({x})=1. Moreover, for such an x we have
(3.14)∑y∈ϕ-1({x})‍μ({y})2μ(ϕ-1({y}))=∑y∈X(i,1)‍μ({y})2μ(ϕ-1({y}))≤∑y∈X(i,1)‍μ({y})2μ(X(i+1,1)).
Therefore, arguing as above we see that the inequality in (3.13) holds for all x∈ℕ×{2} as well.
It is easy to see that the operatorCϕ is densely defined. This follows immediately from Lemma 2.1, formula (3.8), and the fact that μ(X(i,1))<∞ for all i∈ℕ. Since (3.13) holds for every x∈X, by Lemma 2.1(iv) we get hyponormality of Cϕ. Now, we observe that both operators Cϕ2 and Cϕ2 have trivial domains. This is a consequence of the fact that there is no x∈X such that χ{x}∈D(Cϕ2). Indeed, take any x∈X. Then x=xn for some n∈ℕ and hence X(n,2)⊆(ϕ2)-1({x}). By definition, we have χ{x}∉D(Cϕ2). Therefore, by Corollary 2.2, the domain of Cϕ2 is trivial. Since D(Cϕ2)⊆D(Cϕ2) (cf. [4, Theorem 5.1]) we get the claim.


## 4. Final Remarks

(1) It is known that there are closed symmetric operators whose squares possess trivial domains (cf. [24, 25]). Since each symmetric operator is automatically subnormal, one could expect to find a subnormal composition operator with trivial square among symmetric ones. However, this is not the case of the space ℓ2(μ). The reason for this is that for every symmetric composition operator Cϕ acting in ℓ2(μ) the space ⋂n=1∞D(Cϕn) is dense in ℓ2(μ) (cf. [5]).(2) By using [3, Theorem 2.7] one can adapt our example, so as to obtain a  composition operator acting in L2-space over a nonatomic measure space (in fact, the measure could be equivalent to the Lebesgue measure on the unit interval), being hyponormal and having trivial square.(3) A considerable amount of attention has been given to the study ofp-hyponormal operators, also in the context of composition operators (cf. [21, 26–29]). Recall that a bounded operator T is said to be p-hyponormal, p∈(0,∞), if (TT*)p≤(T*T)p. A complete characterization of p-hyponormal composition operators on L2-spaces was given in [21]. It resembles the characterization of hyponormality and states that Cϕ is p-hyponormal if and only if hϕ>0 a.e. [μ] and
(4.1)hϕp∘ϕ·E(1hϕp)⩽1a.e.[μ].
To the best of our knowledge, the notion has not been introduced to the theory of unbounded operators yet. One can use condition (4.1) as a definition of “unbounded” p-hyponormality, of course only for composition operators. Our construction can be adjusted so as to obtain a composition operator satisfying (4.1) for any p≥1 (it suffices to modify (3.2) slightly). It is also worth pointing out that in our setting the “p-hyponormality” condition takes the following form:
(4.2)μ(ϕ-1({x}))p-1μ({x})p∑y∈ϕ-1({x})‍μ({y})p+1μ(ϕ-1({y}))p⩽1,x∈ϕ(X).



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