A Note on Unbounded Hyponormal Composition Operators in 𝐿²-Spaces

Abstract

We construct an unbounded hyponormal composition operator 𝐶_𝜙 in 𝐿²-space such that the domains of 𝐶²_𝜙 and 𝐶_𝜙² are trivial.

1. Introduction

The class of unbounded composition operators in 𝐿²-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 in ℓ²(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 [6]) are no longer valid in the unbounded case (see also [1]).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=1𝐷(𝐶^𝑛_𝜙) 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=1𝐷(𝐶^𝑛_𝜙); 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 𝐶_𝜙 such that ⋂^∞_𝑛=1𝐷(𝐶^𝑛_𝜙) 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 by ℤ and ℕ the sets of integers and positive integers, respectfully. By ℤ₊ we understand the set ℕ∪{0}. If 𝑋 is a set, then 𝜒_𝑋 stands for the characteristic function of 𝑋; the power set of 𝑋 will be denoted by 2^𝑋.

If ℋ 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 in 𝐿²-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 2^𝑋 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 𝐷𝐶_𝜙=𝑓∈ℓ²𝜇∶𝑓∘𝜙∈ℓ²𝜇,𝐶_𝜙𝑓=𝑓∘𝜙,𝑓∈𝐷𝐶_𝜙, (2.1) defines the so-called composition operator𝐶_𝜙∶ℓ²𝜇⊇𝐷𝐶_𝜙⟶ℓ²𝜇. (2.2) It is well known that 𝐶_𝜙 is a closed linear operator in ℓ²(𝜇). Typically, properties of 𝐶_𝜙 are written in terms of the canonical Radon-Nikodym derivative ℎ_𝜙 attached to 𝐶_𝜙 (cf. [2–4, 6, 17–22]). In our setting, ℎ_𝜙 is given by ℎ_𝜙(𝑥)=𝜇𝜙⁻¹({𝑥})𝜇({𝑥}),𝑥∈𝑋. (2.3) The operator of conditional expectation is also frequently employed when investigating composition operators. Let us recall that for a 𝜎-algebra 𝒜⊆2^𝑋 generated by a partition {𝜎_𝜄∶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: 𝐸𝑓∣𝒜=_𝜄∈ℐ1𝜇𝜎_𝜄𝜎𝜄𝑓𝑑𝜇⋅𝜒_𝜎𝜄. (2.4) We refer the reader to [23] for background on conditional expectation.

The following lemma delivers few basic properties of composition operators in ℓ²(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)𝐶_𝜙 is densely defined. (ii)ℰ⊆𝐷(𝐶_𝜙). (iii)𝜇(𝜙⁻¹({𝑥}))<∞ for all 𝑥∈𝑋.
Moreover, if 𝐶_𝜙 is densely defined, then the following assertion holds: (iv)𝐶_𝜙 is hyponormal if and only if 1𝜇({𝑥})_{𝑦∈𝜙−1({𝑥})}𝜇𝑦²𝜇𝜙⁻¹𝑦≤1,𝑥∈𝜙(𝑋). (2.5)

Proof. By definition, ℰ⊆𝐷(𝐶_𝜙) if and only if 𝜒_{𝑥}∘𝜙∈ℓ²(𝜇) for all 𝑥∈𝑋. Since 𝜒_{𝑥}∘𝜙²𝑑𝜇=𝜒_{𝜙−1({𝑥})}²𝑑𝜇=𝜇𝜙⁻¹({𝑥}),𝑥∈𝑋, (2.6) we see that conditions (ii) and (iii) are equivalent.
That (ii) implies (i) is obvious. The converse can be proved as follows. Let 𝑓∈𝐷(𝐶_𝜙). Then 𝑓=∑_{𝑥∈𝑋𝑓}𝑓(𝑥)⋅𝜒_{𝑥}, where 𝑋_𝑓∶={𝑥∈𝑋∶𝑓(𝑥)≠0}. If 𝑥₀∈𝑋_𝑓, then 𝜒_{𝑥0}∈𝐷(𝐶_𝜙) because of the following 𝑓𝑥₀²𝜒_{𝑥0}∘𝜙𝑑𝜇=𝑓∘𝜙²⋅𝜒_{𝜙−1({𝑥0})}𝑑𝜇≤𝑓∘𝜙²𝑑𝜇<∞. (2.7) If 𝐷(𝐶_𝜙)=ℓ²(𝜇), then clearly 𝑋=⋃_{𝑓∈𝐷(𝐶𝜙)}𝑋_𝑓. Therefore, 𝜒_{𝑥}∈𝐷(𝐶_𝜙) 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]): 𝐶_𝜙 is hyponormal if and only if ℎ_𝜙1∘𝜙⋅𝐸ℎ_𝜙≤1; (2.8) 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 in ℓ²(𝜇) is trivial.

Corollary 2.2. 𝐷(𝐶_𝜙)={0} if and only if ℰ∩𝐷(𝐶_𝜙)=∅.

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 set 𝑋, a positive measure 𝜇 on 2^𝑋 and a map 𝜙∶𝑋→𝑋 such that the composition operator 𝐶_𝜙 acting in ℓ²(𝜇) is hyponormal and 𝐷(𝐶_𝜙²)=𝐷(𝐶²_𝜙)={0}.

Proof. Set 𝑋=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: 𝑋()=),(),()},𝑋()=),(),(),()},𝑋()=),(),(),()},𝑋()=),(),(),()},𝑋()=),(),()}.⋮1,𝑘{(3,𝑘7,𝑘12,𝑘,…2,𝑘{(1,𝑘4,𝑘8,𝑘13,𝑘,…3,𝑘{(2,𝑘5,𝑘9,𝑘14,𝑘,…4,𝑘{(6,𝑘10,𝑘15,𝑘21,𝑘,…5,𝑘{(11,𝑘16,𝑘22,𝑘,… (3.1) Then the sets 𝑋(𝑛,𝑘) are pairwise disjoint, infinite, and 𝑋=⋃_{𝑛∈ℕ,𝑘=1,2}𝑋(𝑛,𝑘).
Now, we define an atomic measure 𝜇 on 𝑋. For this we take any sequence {𝑎(𝑛)}^∞_𝑛=1 of positive real numbers such that 2¹¹≤2^𝑘+2(𝑎(𝑘))²≤𝑎(𝑘+1),𝑘∈ℕ. (3.2) Clearly, {𝑎(𝑛)}^∞_𝑛=1 is monotonically increasing and 𝑎(𝑘)⩾2^𝑘+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=(𝑘_𝑖,1). Then we put 𝜇𝑥=𝑎(𝑛)({})2^𝑖. (3.3) On the other hand, if 𝑥∈ℕ×{2}, then we set 𝜇({𝑥})=1. (3.4) Since 𝑋=⋃_{𝑛∈ℕ,𝑘=1,2}𝑋(𝑛,𝑘), the measure 𝜇 is defined on the whole of 2^𝑋. 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={𝐱_𝑛∶𝑛∈ℕ}, where 𝐱_𝑛(),∶=1,2if(),𝑛=1,𝑙+1,2if(),𝑛=2𝑙,𝑙∈ℕ,𝑛∈ℕ,𝑙,1if𝑛=2𝑙+1,𝑙∈ℕ. (3.5) It is easy to see that 𝐱_𝑚∉𝑋(𝑚,1) for 𝑚∈ℕ. Moreover, if 𝐱_𝑖∈𝑋(𝑚,1) for some 𝑖,𝑚∈ℕ, then 𝑖>𝑚. Now, if 𝑥∈𝑋(𝑛,1) for some 𝑛∈ℕ, then we set 𝜙(𝑥)=𝐱_𝑛, (3.6) if 𝑥=(𝑚,2) for some 𝑚∈ℕ, then we put 𝜙(𝑥)=(𝑚,1). (3.7) Again, equality 𝑋=⋃_{𝑛∈ℕ,𝑘=1,2}𝑋(𝑛,𝑘) yields that the transformation 𝜙 is defined on the whole of 𝑋.
Now, we derive some complementary information on 𝜙 and 𝜇. Choose 𝑥∈𝑋. Then 𝑥=(𝑝,𝑘)∈𝑋(𝑛,𝑘) for some 𝑝,𝑛∈ℕ and 𝑘∈{1,2}. Simultaneously, 𝑥=𝐱_𝑖 for some 𝑖∈ℕ. Therefore, we gather that 𝜙⁻¹𝑥=𝑋(),({})𝑖,1for𝑋()∪,𝑘=2,𝑖,1𝑝,2for𝑘=1. (3.8) Furthermore, if 𝑦∈𝜙⁻¹({𝑥}), then either 𝑦=(𝑝,2) or 𝑦=(𝑟,1) for some 𝑟∈ℕ. In both cases 𝑦=𝐱_𝑚 for some 𝑚∈ℕ. This yields 𝜙⁻¹𝑦=𝑋(𝑚,1)∪{(𝑟,2)},for𝑋(),𝑘=2,𝑚,1for,𝑋()∪,𝑘=1,𝑦=𝑝,2𝑚,1{(𝑟,2)}for𝑘=1,𝑦≠𝑝,2. (3.9) Suppose now that 𝑥∈𝑋(𝑛,1). Then, as above, 𝑥=(𝑝,1)=𝐱_𝑖 for 𝑝,𝑖∈ℕ. Note that 𝜇𝑥≥𝑎(𝑛)({})2^𝑖≥12^𝑖. (3.10) Also, if 𝑦=𝐱_𝑙∈𝜙⁻¹({𝑥})⧵{(𝑝,2)}, then 𝑙>𝑖. This, (3.8), (3.9), definition of 𝜇, and (3.2), implies that _{𝑦∈𝜙−1({𝑥})}𝜇𝑦²𝜇𝜙⁻¹𝑦=𝜇𝑝,2²𝜇𝜙⁻¹𝑝,2+_{𝑦∈𝑋(𝑖,1)}𝜇𝑦²𝜇𝜙⁻¹𝑦⩽1𝜇𝑋𝑚^′,1+_{𝑦∈𝑋(𝑖,1)}𝜇𝑦²𝜇(𝑋(⩽1𝑖+1,1))2^𝑚′+4+𝜇(𝑋(𝑖,1))²𝜇(𝑋(≤1𝑖+1,1))2^𝑚′+4+12^𝑖+1, (3.11) where 𝑚^′∈ℕ obeys (𝑝,2)=𝐱_𝑚′. For such an 𝑚^′ we either have 𝑖=𝑚^′+2 (for 𝑝=1) or 𝑖=𝑚^′+3 (for 𝑝>1), thus we obtain _{𝑦∈𝜙−1({𝑥})}𝜇𝑦²𝜇𝜙⁻¹𝑦≤12^𝑖. (3.12) This together with (3.10) implies that for all 𝑥∈⋃^∞_𝑛=1𝑋(𝑛,1) we have _{𝑦∈𝜙−1({𝑥})}𝜇𝑦²𝜇𝜙⁻¹𝑦≤𝜇({𝑥}). (3.13) On the other hand, if 𝑥=𝐱_𝑖∈𝑋(𝑛,2), then 𝜇({𝑥})=1. Moreover, for such an 𝑥 we have _{𝑦∈𝜙−1({𝑥})}𝜇𝑦²𝜇𝜙⁻¹𝑦=_{𝑦∈𝑋(𝑖,1)}𝜇𝑦²𝜇𝜙⁻¹𝑦≤_{𝑦∈𝑋(𝑖,1)}𝜇𝑦²𝜇(𝑋(.𝑖+1,1)) (3.14) 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 operator 𝐶_𝜙 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 𝐶_𝜙. Now, we observe that both operators 𝐶_𝜙² and 𝐶²_𝜙 have trivial domains. This is a consequence of the fact that there is no 𝑥∈𝑋 such that 𝜒_{𝑥}∈𝐷(𝐶_𝜙²). Indeed, take any 𝑥∈𝑋. Then 𝑥=𝐱_𝑛 for some 𝑛∈ℕ and hence 𝑋(𝑛,2)⊆(𝜙²)⁻¹({𝑥}). By definition, we have 𝜒_{𝑥}∉𝐷(𝐶_𝜙²). Therefore, by Corollary 2.2, the domain of 𝐶_𝜙² is trivial. Since 𝐷(𝐶²_𝜙)⊆𝐷(𝐶_𝜙²) (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 𝐶_𝜙 acting in ℓ²(𝜇) the space ⋂^∞_𝑛=1𝐷(𝐶^𝑛_𝜙) 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 of 𝑝-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 𝐶_𝜙 is 𝑝-hyponormal if and only if ℎ_𝜙>0 a.e. [𝜇] and ℎ^𝑝_𝜙1∘𝜙⋅𝐸ℎ^𝑝_𝜙⩽1a.e.𝜇. (4.1) 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: 𝜇𝜙⁻¹({𝑥})^𝑝−1𝜇({𝑥})^𝑝_{𝑦∈𝜙−1({𝑥})}𝜇𝑦^𝑝+1𝜇𝜙⁻¹𝑦^𝑝⩽1,𝑥∈𝜙(𝑋). (4.2)

Acknowledgments

The authors would like to thank both referees for pointing out an error in earlier version of the paper and for valuable comments that helped to improve the presentation. The research was supported by the NCN (National Science Center) Grant DEC-2011/01/D/ST1/05805. This paper is dedicated to Professor J. Stochel on the occasion of his sixtieth birthday.