Abstract
In this paper, we investigate topological transitivity of operators on nonseparable Hilbert spaces which are similar to backward weighted shifts. In particular, we show that abstract differential operators and dual operators to operators of multiplication in graded Hilbert spaces are similar to backward weighted shift operators.
1. Introduction
Let be a Hausdorff locally convex space. A continuous operator is called topologically transitive if for each pair of nonempty open subsets of , there is some with If the underlying space is a separable Baire space, the transitivity is equivalent to the hypercyclicity by Birkhoff’s transitivity theorem (see [1], p. 2). A continuous linear operator acting on a separable Fréchet space is called hypercyclic if there is a vector for which the orbit under is dense in Every such vector is called a hypercyclic vector of
The study of topological transitivity was started from Birkhoff’s result [2] in 1929 for nonlinear continuous functions. This result is important in the theory of chaos (see e.g., [3]). For the space of all entire functions, Birkhoff [2] also proved that the translation operator is hypercyclic. MacLane in 1952 showed that the differentiation operator is hypercyclic too (see [4]).
A lot of work on hypercyclicity has been based on the well-known so-called hypercyclicity criterion (the Kitai-Gethner-Shapiro theorem). In [5], Bermúdez and Kalton observed that similar criterion holds for topologically transitive operators on Banach spaces.
Theorem 1 (topologically transitive criterion). Let be a bounded linear operator on a complex Banach space (not necessarily separable). Suppose that there exists a strictly increasing sequence of positive integers for which there are (i)A dense subset such that for every as (ii)A dense subset and a sequence of mappings such that for every and for every as Then, is topologically transitive.
More information on transitive and hypercyclic operators can be found in [1, 6]. The existence of an element with a dense orbit implies that must be separable. On the other hand, the transitivity of does not require that the space is separable.
Problem 2 (Bermúdez and Kalton [5]). Is there any characterization of nonseparable Banach spaces which support a topologically transitive operator?
Note that any hypercyclic operator always has an invariant, norm dense, linear subspace in which every nonzero vector is a hypercyclic vector for
Theorem 3 (see [7]). Let be a complete metric space which has no isolated points. Let be topologically transitive operator and There exists a dense subset of such that for each , there exists a -invariant (separable) closed subspace of with such that the restriction of to is hypercyclic.
The hypercyclisity of composition operators and differentiation operators on some function spaces of finite and infinite variables was studied in [8–14]. According to a classical theorem of Rolewicz [15], the weighted backward shifts on is hypercyclic if This fact remains to be true if we replace into for Bermúdez and Kalton in [5] showed that spaces like and ) do not support topologically transitive operators, where is the space of all bounded operators on
In [7] (Proposition 3.4), Manoussos proved that the backward weighted shift is topologically transitive on nonseparable Hilbert space. More detailed, let be a Hilbert space (possibly nonseparable) and be the -sum of infinitely many copies of That is, if then
Then, the backward weighted shiftwith weight sequence is defined by the following:
Unfortunately, Proposition 3.4 in [7] contains a slight inaccuracy which for us is essential. So, we propose the corrected version.
Proposition 4. Let be a Hilbert space and be a backward weighted shift with positive weight sequence . The following are equivalent: (i) is topologically transitive(ii)There exists a nontrivial -invariant (separable) closed subspace on which the restriction of to is hypercyclic(iii)The restriction to any -invariant (separable) closed subspace which contains nonzero vectors of the form for every is hypercyclic(iv)
Note that in [7], item (iii) is written: “The restriction to any -invariant (separable) closed subspace is hypercyclic.” But it is not correct because the subspace consisting of vectors is invariant but the restriction of to any separable subspace of is not hypercyclic since
In Section 2, we consider abstract shift similar operators on nonseparable function Hilbert spaces In particular, abstract differentiation operators and dual operators to abstract multiplication operators can be considered as abstract shift similar operators. In Section 3, we construct examples of topologically transitive operators.
2. Abstract Shift Similar Operators
Let be a sequence of Hilbert spaces. Throughout this paper, we assume that all are nontrivial; that is, and not necessary separable. Let us suppose that for every and is isomorphic to We denote by the Hilbert space consisting of elements endowed with norm Let be a sequence of positive numbers (weights). Let us fix a sequence of isomorphisms An operator will be called a backward weighted shift (with respect to the family) with weight sequence if it is of the following form:
From Proposition 4, it follows the next corollary.
Corollary 5. Let be a sequence of Hilbert spaces and be a backward weighted shift with respect to and with positive weight sequence . Let us suppose that Then, the following are equivalent: (i) is topologically transitive(ii)There exists a nontrivial -invariant (separable) closed subspace on which the restriction of to is hypercyclic(iii)The restriction to any -invariant (separable) closed subspace which contains nonzero vectors of the form for every is hypercyclic(iv)
Proof. We set Then,
and so on. For any Because of Equation (6), the inverse operator
is well defined and bounded. So, is an isometric isomorphism from to For a closed subspace the range is a closed subspace of Let Then, is a backward weighted shift on with positive weight sequence . Since is an isomorphism, the topological transitivity of or hypercyclicity of on is equivalent to the topological transitivity of on or hypercyclicity of on respectively. Hence, we can apply Theorem 3 for
Note that the case when and can be reduced to our case if we consider and instead of and respectively.
An operator is a backward weighted shift similar operator if there exists an isomorphism for some sequence of Hilbert spaces and a backward weighted shift such that It is clear that is topologically transitive if and only if is topologically transitive.
Let us suppose that and the isomorphisms be such that is an isometric isomorphism of onto and maps onto Also, we denote by the projection of onto Then, we define a backward weighted partial shift (with respect to family ) with weight sequence on by the following:
In other words, and is a backward weighted shift on and
Let us recall that the density of a metric space is the smallest cardinality of a dense subset of It is well known that two infinite dimensional Hilbert spaces and are isometrically isomorphic if and only if The next proposition shows that the backward weighted partial shift can be reduced to the weighted shift.
Proposition 6. Let be a backward weighted partial shift with respect to the family of operators with weight sequence on Let us suppose that all are infinite dimensional for and Then, is a backward weighted shift similar operator; that is, there is a sequence of Hilbert spaces and an isometric isomorphism such that is a backward weighted shift.
Proof. Let us define spaces by the following way: Since spaces are isomorphic each to others, for all Since we have and so and are isomorphic for all It is clear that and are just different representations of the same space, and so is the identical operator. Also, the restrictions of to are isometric isomorphisms. So, is a backward weighted shift with weight sequence on
Corollary 7. Let be a backward weighted partial shift with positive weight sequence which satisfies conditions of Proposition 6. Then, the following are equivalent: (i) is topologically transitive(ii)There exists a nontrivial -invariant (separable) closed subspace on which the restriction of to is hypercyclic(iii)The restriction to any -invariant (separable) closed subspace which contains nonzero vectors of the form for every is hypercyclic(iv)
This approach can be generalised by the following way.
Theorem 8. Let be a surjective bounded operator on a -sum of Hilbert spaces such that Then, is a backward weighted shift similar operator with some weight sequence where is the restriction of to
Proof. Let us define the following spaces:
Let us denote by the following:
In other words, if then Let be the restriction of to Then, Thus, the action of on can be written by the following:
and So, It is clear that the spaces and consist of the same vectors and the identity map is an isometric isomorphism. Thus, is the restriction of to By the construction, and Moreover, since if In addition, since is onto, must be onto for every because the preimage of under is Hence, every is an isomorphism.
Let us set the following:
Then for every that is, is a backward weighted shift.
Let us show that for every Since all spaces are nontrivial and is onto, there is a vector such that So, and ; that is, If already we have nonzero elements then we denote by a vector in in the preimage of under By the assumption so Thus, and Hence,
An element is a finite type vector if there is such that for all
Definition 9. Let be the Hilbert spaces. We say that is a graded Hilbert space, if there is a bilinear map (multiplication) defined for every finite type vector and an arbitrary with values in such that the multiplication is associative, nondegenerated (that is, only if or ) and for every and
Definition 10. For a given graded Hilbert space and a finite type vector , we denote by the multiplication operator.
Theorem 11. Let and be continuous on If the dual operator is surjective, then it is a backward weighted shift similar operator.
Proof. Let
and be the Hermitian dual space for every Then, maps to and maps to and Let be the restriction of to Since is continuous, is continuous on as well and We set the following:
Since is surjective, are surjective for all and since are bijective for So,
is a backward weighted shift on
Note that if then and where is the restriction of to
Definition 12. Let be a graded Hilbert space. A continuous linear operator is called a differential operator if the restriction of to maps onto and for every finite type vector and for all
Corollary 13. Let be a differential operator on a graded Hilbert space such that is onto and Then, there is a sequence of Hilbert spaces and an isometric isomorphism such that is a backward weighted shift with some weight sequence
Proof. Since there is such that So, Moreover, for every and Thus, and we can apply Theorem 2.4.
Corollary 14. Let be a continuous surjective differential operator on a graded Hilbert space The following are equivalent: (i) is topologically transitive(ii)There exist and such that and(iii)
Proof. As we observed, and so By Theorem 8., So if for all then and must be topologically transitive.
3. Examples of Topologically Transitive Operators
3.1. Partial Shifts
It is well known that any element of the (complete) Hilbert tensor product can be represented by the following series or by the following infinite matrix with Note that can be considered as by the following representation: where
.
Let be an infinite matrix of positive numbers. We consider the following operators on Let and be the transposed operator, Let From Theorem 8., it follows that if is continuous and
then operators are topologically transitive for every In particular, the operator looks like weighted diagonal shift operator. It is clear that these examples can be generalised for any tensor degree Next, we consider a nonseparable case.
Let be the set of indexes (uncountable, in general) which can be represented as a disjoint union of subsets . and there are a sequence of bijections So, we can write and define by By Corollary 5, we have that is topologically transitive if and only if the weight satisfies Equation (31). Moreover, the Hilbert tensor product is isomorphic to For a given weight matrix , we define operators and on by the following:
where
Corollary 15. Let then for every , operators are topologically transitive if and only if
3.2. Differential Operator on a Function Space
Examples of Hilbert spaces of analytic functions with continuous differential operators can be found among reproducing kernel Hilbert spaces.
Definition 16. Let be an abstract set and a Hilbert space of complex valued functions on equipped with inner product A function defined on is called reproducing kernel of a closed subspace if
(i)For any fixed belongs to as a function in (ii)For any and for any The space is called a Hilbert space with reproducing kernel. A function such that
for every and is called the kernel function of
Let be the Hilbert space of power series
endowed with the norm
where is a positive sequence and So, the inner product on is defined by and is a reproducing kernel space with the reproducing kernel.
It is known [16] that if as consists of entire functions on
Let be the differential operator on Then, , and so . Thus, is continuous (c. f. [9]) if and only if
On the other hand, according to Corollary 14, is topologically transitive and so hypercyclic if and only if
It is clear that Let us consider the multiplication operator on Then, According to Theorem 11, is bounded if and only if and is topologically transitive if and only if
This result agrees with [8]. Various Hilbert spaces of analytic functions of infinite many variables were constructed in [14, 17, 18] using the Hilbertian symmetric tensor products of Hilbert spaces. Such constructions can be easily extended to nonseparable Hilbert spaces. In [14], the authors found some conditions under which the translation operator on a Hilbert space of entire analytic functions on is hypercyclic. Taking into account that the translation operator can be represented as an exponential function of a differential operator, we would like to ask the following question.
Question 17. Let be a differential operator on a graded nonseparable Hilbert space. Suppose that is topologically transitive. Does topologically transitive?
4. Conclusion
We can see that dual to abstract multiplication operators and abstract differentiation operators on Hilbert spaces can be represented as backward weighted shift operators. Thus, we can use known results about topological transitivity of backward weighted shift operators to get conditions of topological transitivity of operators and
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research was funded by the National Research Foundation of Ukraine (2020.02/0025, 0121U111037).