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A Note on Unbounded Hyponormal Composition Operators in -Spaces
We construct an unbounded hyponormal composition operator in -space such that the domains of and are trivial.
The class of unbounded composition operators in -spaces seems to be an interesting object of research. It contains some classical operators (e.g., the shift operator on weighted 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 ). 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 whose square has trivial domain. Existence of such example is relevant to the problem of characterizing subnormality of unbounded composition operators in -spaces. In a recent paper , it was shown that the celebrated Lambert's characterizations of subnormality for bounded composition operators (cf. ) are no longer valid in the unbounded case (see also ). The authors also elaborated on the role of density of in the underlying -space. It is important because of all known general criteria for subnormality of unbounded operators which do not require the density of ; 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 such that is not dense in the underlying -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. ). As it turned out very recently (cf. [1, Lemma ]), some weighted shifts on directed trees are unitarily equivalent to composition operators in -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.
We denote by and the sets of integers and positive integers, respectfully. By we understand the set . If is a set, then stands for the characteristic function of ; the power set of will be denoted by .
If is a (complex) Hilbert space and is a (linear) operator acting in , then stands for the domain of and stands for the adjoint operator of (if it exists). A densely defined operator in is said to be hyponormal if and for all . 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 -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 be a countably infinite set, let be a measure on such that for every , and let . By we denote the Hilbert space of all complex valued functions on which are square-summable with respect to the measure . The following formula defines the so-called composition operator It is well known that is a closed linear operator in . Typically, properties of are written in terms of the canonical Radon-Nikodym derivative attached to (cf. [2–4, 6, 17–22]). In our setting, is given by The operator of conditional expectation is also frequently employed when investigating composition operators. Let us recall that for a -algebra generated by a partition of and for a nonnegative function on , the conditional expectation of with respect to can be defined as follows: We refer the reader to  for background on conditional expectation.
The following lemma delivers few basic properties of composition operators in 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: stands for the set .
Lemma 2.1. The following conditions are equivalent(i) is densely defined. (ii). (iii) for all .
Moreover, if is densely defined, then the following assertion holds: (iv) is hyponormal if and only if
Proof. By definition, if and only if for all . Since
we see that conditions (ii) and (iii) are equivalent.
That (ii) implies (i) is obvious. The converse can be proved as follows. Let . Then , where . If , then because of the following If , then clearly . Therefore, for every , 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 (see [2, Corollary 6.7]): is hyponormal if and only if the conditional expectation above is calculated with respect to the completion of the -subalgebra of . In our situation the -subalgebra is generated by the partition . 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 is trivial.
Corollary 2.2. if and only if .
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 , a positive measure on and a map such that the composition operator acting in is hyponormal and .
Proof. Set . Let be a family of subsets of defined according to the following scheme:
Then the sets are pairwise disjoint, infinite, and .
Now, we define an atomic measure on . For this we take any sequence of positive real numbers such that Clearly, is monotonically increasing and for all . Suppose for some . Since , where are nonnegative integers, there exists such that . Then we put On the other hand, if , then we set Since , the measure is defined on the whole of . Observe that for every . In order to define a transformation of , we note that , where It is easy to see that for . Moreover, if for some , then . Now, if for some , then we set if for some , then we put Again, equality yields that the transformation is defined on the whole of .
Now, we derive some complementary information on and . Choose . Then for some and . Simultaneously, for some . Therefore, we gather that Furthermore, if , then either or for some . In both cases for some . This yields Suppose now that . Then, as above, for . Note that Also, if , then . This, (3.8), (3.9), definition of , and (3.2), implies that where obeys . For such an we either have (for ) or (for ), thus we obtain This together with (3.10) implies that for all we have On the other hand, if , then . Moreover, for such an we have Therefore, arguing as above we see that the inequality in (3.13) holds for all as well.
It is easy to see that the operator is densely defined. This follows immediately from Lemma 2.1, formula (3.8), and the fact that for all . Since (3.13) holds for every , by Lemma 2.1(iv) we get hyponormality of . Now, we observe that both operators and have trivial domains. This is a consequence of the fact that there is no such that . Indeed, take any . Then for some and hence . By definition, we have . Therefore, by Corollary 2.2, the domain of is trivial. Since (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 . The reason for this is that for every symmetric composition operator acting in the space is dense in (cf. ).
(2) By using [3, Theorem 2.7] one can adapt our example, so as to obtain a composition operator acting in -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 -hyponormal operators, also in the context of composition operators (cf. [21, 26–29]). Recall that a bounded operator is said to be -hyponormal, , if . A complete characterization of -hyponormal composition operators on -spaces was given in . It resembles the characterization of hyponormality and states that is -hyponormal if and only if a.e. and 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” -hyponormality, of course only for composition operators. Our construction can be adjusted so as to obtain a composition operator satisfying (4.1) for any (it suffices to modify (3.2) slightly). It is also worth pointing out that in our setting the “-hyponormality” condition takes the following form:
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.
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