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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 791817, 6 pages
Subnormal Weighted Shifts on Directed Trees and Composition Operators in 2-Spaces with Nondensely Defined Powers
1Katedra Zastosowań Matematyki, Uniwersytet Rolniczy w Krakowie, Ulica Balicka 253c, 30-198 Kraków, Poland
2Instytut Matematyki, Uniwersytet Jagielloński, Ulica Łojasiewicza 6, 30-348 Kraków, Poland
Received 15 October 2013; Accepted 3 December 2013; Published 19 February 2014
Academic Editor: Henryk Hudzik
Copyright © 2014 Piotr Budzyński et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 ()th power is not. As a consequence, for every positive integer n there exists a nonsymmetric subnormal composition operator C in an L2-space over a σ-finite measure space such that Cn is densely defined and is not.
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 ; see also  for a different construction). More than four decades later, Schmüdgen discovered another pathological behaviour of domains of powers of symmetric operators (cf. ). It is well known that symmetric operators are subnormal (cf. [4, Theorem 1 in Appendix ]). 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. ). In the present paper we discuss the above question in the context of subnormal weighted shifts on directed trees and subnormal composition operators in -spaces (over -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. ). The same applies to symmetric composition operators in -spaces (cf. [9, Proposition ]). Formally normal composition operators in -spaces, which may be unbounded (see [9, Appendix ]), 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 there exists a subnormal weighted shift on a directed tree whose th power is densely defined while its th power is not. A similar question can be asked for composition operators in -spaces. To answer both of them, we proceed as follows. First, by applying a recently established criterion for subnormality of weighted composition operators in -spaces which makes no appeal to density of -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 of Borel probability measures on which satisfies (6)) is subnormal and, what is more, its th power is densely defined if and only if all moments of these measures up to degree 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, , , , , and stand for the sets of integers, nonnegative integers, positive integers, nonnegative real numbers and complex numbers, respectively. Set . We write for the -algebra of all Borel subsets of . Given , we denote by the Borel probability measure on concentrated on .
The domain of an operator in a complex Hilbert space is denoted by (all operators considered in this paper are linear). Set . Recall that a closed densely defined operator in is said to be normal if (see [11, 14, 15] for more on this class of operators). We say that a densely defined operator in is subnormal if there exist a complex Hilbert space and a normal operator in such that (isometric embedding) and for all . 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 is a -finite measure space, is an -measurable function, and is an -measurable mapping. Define the -finite measure by for . Let be the measure given by for . Assume that is absolutely continuous with respect to . By the Radon-Nikodym theorem (cf. [20, Theorem ]), there exists a unique (up to a.e. equivalence) -measurable function such that Then the operator in , given by is well defined (cf. [21, Proposition 7]). Call a weighted composition operator. By [21, Proposition 10], is densely defined if and only if a.e. ; moreover, if this is the case, then is -finite and, by the Radon-Nikodym theorem, for every -measurable function there exists a unique (up to a.e. equivalence) -measurable function such that We call the conditional expectation of with respect to (see  for more information). A mapping is called an -measurable family of probability measures if the set-function is a probability measure for every and the function is -measurable for every .
The following criterion (read: a sufficient condition) for subnormality of unbounded weighted composition operators is extracted from [21, Theorem 29].
Theorem 1. If is densely defined, a.e. , and there exists an -measurable family of probability measures such that then is subnormal.
3. Weighted Shifts on Directed Trees
Let be a directed tree ( and stand for the sets of vertices and edges of , resp.). Set for . Denote by the partial function from to which assigns to a vertex its parent (i.e., a unique such that ). A vertex is called a root of if has no parent. A root is unique (provided it exists); we denote it by . Set if has a root and otherwise. We say that is a branching vertex of and write , if consists of at least two vertices. We refer the reader to  for all facts about directed trees needed in this paper.
By a weighted shift on with weights we mean the operator in defined by where is the mapping defined on functions via As usual, is the Hilbert space of square summable complex functions on with standard inner product. For , we define to be the characteristic function of the one-point set . Then is an orthonormal basis of .
The following useful lemma is an extension of part (iv) of [13, Theorem ].
Lemma 2. Let be a weighted shift on a directed tree with weights and let . Then is densely defined if and only if for every .
Proof. In view of [13, Theorem (iv)], is densely defined if and only if for every . Note that if and , then and , which implies that whenever . In turn, if , then clearly . Using the above and an induction argument (related to paths in ), we deduce that is densely defined if and only if for every .
It is worth mentioning that if , then, by Lemma 2 and [13, Theorem (iv)] (or by the proof of Lemma 2), is dense in . 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 ], we do not assume the density of -vectors in the underlying -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 ]). Here, and later, we adopt the conventions that , and ; we also write in place of .
Theorem 3. Let be a weighted shift on a countably infinite directed tree with weights . Suppose that there exist a system of Borel probability measures on and a system of nonnegative real numbers such that Then the following two assertions hold:(i)if is densely defined, then is subnormal,(ii)if , then is densely defined if and only if for all .
Proof. (i) Assume that is densely defined. Set and . Let be the counting measure on ( is -finite because is countable). Define the weight function and the mapping by
Clearly, the measure is absolutely continuous with respect to and
Thus, by [12, Proposition ], for every . We claim that a.e. . This is the same as to show that if and , then . Thus, if and , then applying (6) to , we deduce that ; in turn, applying (6) to with , we get , which proves our claim.
Note that (the disjoint union). Hence, the conditional expectation of a function with respect to is given by where (see also (8)); on the remaining part of we can put .
Substituting into (6), we see that for every such that . Thus, using the standard measure-theoretic argument and (6), we deduce that Set for and . It follows from (9) and (10) that : is a (-measurable) family of probability measures which fulfils the following equality: This implies that satisfies . Hence, by Theorem 1, the weighted composition operator (see (2)) is subnormal. Since , assertion (i) is proved.
(ii) It is easily seen that if is a finite positive Borel measure on and for some , then for every such that . This fact combined with Lemma 2 and [22, Lemmata (i) and (i)] implies assertion (ii).
Remark 4. Assume that is a densely defined weighted shift on a countably infinite directed tree with weights . A careful inspection of the proof of Theorem 3 reveals that if (with ) is a system of Borel probability measures on which satisfies (6), then a.e. , the family defined by for satisfies , and for every . We claim that if a.e. and is any family of probability measures which satisfies , then the system of probability measures defined by satisfies (6) with in place of . Indeed, implies (11). Hence, by (9), equality in (10) holds for every with for . This implies via the standard measure-theoretic argument that equality in (6) holds for every . Since a.e. , we deduce that equality in (6) holds for every with in place of . Clearly, this is also the case for . 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 with , we define the directed tree as follows (the symbol “” denotes disjoint union of sets): where for . Clearly, is leafless and is its only branching vertex. From now on, we write instead of the more formal expression whenever .
Theorem 5. Let be such that and let be a weighted shift on a directed tree with nonzero weights . Suppose that there exists a sequence of Borel probability measures on such that
and that one of the following three disjunctive conditions is satisfied: (i) and
(iii) and equalities (16) and (17) are valid.
Then the following two assertions hold: (a)if is densely defined, then is subnormal,(b)if , then is densely defined if and only if
Proof. As in the proof of [23, Theorem 4.1], we define the system of Borel probability measures on and verify that satisfies (6). Hence, assertion (a) is a direct consequence of Theorem 3(i).
(b) Fix . It follows from Theorem 3(ii) that is densely defined if and only if . Using the explicit definition of and applying the standard measure-theoretic argument, we see that This completes the proof of assertion (b) (the case of can also be settled without using the definition of simply by applying Lemma 2 and [12, Proposition (iii)]).
5. The Example
It follows from [22, Lemma (i)] that if is a weighted shift on and , then is dense in (this means that Corollary 6 is interesting only if ). If , the situation is completely different. Using Theorem 5 and Corollary 6, we show that for every and for every , there exists a subnormal weighted shift on such that is densely defined and is not. For this purpose, we adapt [12, Procedure ] to the present context. In the original procedure, one starts with a sequence of Borel probability measures on (whose th moments are finite for every such that ) and then constructs a system of nonzero weights 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 are measures with two-point supports (this question is not discussed here).
Example 1. Assume that . Consider the measures with for . By [12, Notation and Procedure ], if and only if . Hence, there is no loss of generality in assuming that . To cover all possible choices of , we look for a system of nonzero weights which satisfies (14), (16), (17) with , (19) and the equality . Setting for , we reduce our problem to find a sequence such that
Indeed, if is such a sequence, then multiplying its terms by an appropriate positive constant, we may assume that satisfies (21) and (16). Next we define the weights recursively so as to satisfy (17) with , and finally we set for all such that . The so constructed weights meet our requirements.
The following lemma turns out to be helpful when solving the reduced problem.
Lemma 7. If is an infinite matrix with entries , then there exists a sequence such that
Proof. First observe that, for every , there exists such that . Hence, for every .
Since , there exists a subsequence of the sequence such that for every . Set . By Lemma 7, there exists such that Define the system by Since for all , we get 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 , then any weighted shift on with nonzero weights is unitarily equivalent to an injective composition operator in an -space over a -finite measure space (cf. [13, Lemma ]). This fact combined with Example 1 shows that, for every , there exists a subnormal composition operator in an -space over a -finite measure space such that is densely defined and is not.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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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