Research Article | Open Access
Leila Bagheri, Bahmann Yousefi, "Cyclicity of Special Operators on a BK with AK Space", International Journal of Analysis, vol. 2015, Article ID 301983, 3 pages, 2015. https://doi.org/10.1155/2015/301983
Cyclicity of Special Operators on a BK with AK Space
Let be a complex domain and let be a reflexive BK space with AK such that and the functional of evaluation at is bounded for all . We will investigate the cyclicity for the adjoint of a weighted composition operator acting on .
We write for the set of all complex sequences . Let denote the set of all finite sequences. By , we denote the sequence with and whenever . For any sequence , let be its -section. Given any subset of , we write for the set of all formal power series with where , regardless of whether or not the series converges for any value of . If endowed with the norm of , then and are norm isomorphic. Let be defined by , so the corresponding shift operator is defined by if and else.
A BK space is a Banach sequence space with the property that convergence implies coordinatewise convergence. A BK space containing is said to have AK if every sequence has a unique representation ; that is, ; it is said to have AD, if is dense in . Given any subset of , the set is called the -dual of .
If is a complex number, then denotes the functional of evaluation at , defined on the polynomials by A point is said to be a bounded point evaluation on if the functional extends to a continuous linear functional on . Finally, we consider the multiplication of formal power series given by where for all integers . If and is a polynomial, then to the vector the formal power series corresponds.
Let be a complex domain and let be a Banach space of formal power series with coefficients in a reflexive BK space with AK such that . It is convenient and helpful to introduce the notation to stand for , for and . We assume and the operators and the functional of evaluation at () are bounded on .
A complex valued function on for which for every is called a multiplier of and the collection of all of these multipliers is denoted by . If is a bounded operator on , the adjoint satisfies In general each multiplier of determines a multiplication operator defined by , . Also It is well-known that each multiplier is a bounded analytic function on . Indeed for each in .
Let be an analytic self-map of the open unit disk . A composition operator maps an analytic function into . If and is bounded, is called a weighted composition operator on . By we will mean the th iterate of .
Let be a Banach space. We denote by the set of bounded operators on the Banach space . Let and . We say that is a cyclic vector of if is equal to the closed linear span of the set An operator is called cyclic if it has a cyclic vector. In this paper, we investigate the cyclicity of weighted composition operators on some BK spaces with AK.
2. Main Results
The sequence spaces has been the focus of attention for several decades and many properties of operators on these spaces have been studied (e.g., ). For , a sequence with for all , Malkowsky considered the space and studied its -dual and characterized some linear operators on . In  the reflexivity of -summable sequences from a Banach space is investigated whenever is a Banach perfect sequence space. Some BK spaces including the spaces and have been introduced in  and also their duals have been computed. In , Aydin and Basar have introduced new classes of sequence spaces which include the spaces and , and the characterization of some other classes of sequence spaces have also been derived. Malafosse has given some properties and applications of Banach algebras of bounded operators , when is a BK space . In , Mursaleen and Noman introduced some spaces of difference sequences which are the BK spaces of nonabsolute type and proved that these spaces are linearly isomorphic to the spaces and , respectively. In [8, 9], Mursaleen and Noman established some identities or estimates for the operator norms and the Hausdorff measures of noncompactness of certain operators on some BK spaces of weighted means have been investigated. Furthermore, by using the Hausdorff measure of noncompactness, they applied their results to characterize some classes of compact operators on those spaces. In  some identities or estimates for the operator norms and the Hausdorff measure of noncompactnesss of certain matrix operators on some BK spaces have been established. In , Basarir and Kara have characterized some classes of compact operators on special normed Riesz sequence spaces by using the Hausdorff measure of noncompactness. A characterization of compact operators between certain BK spaces has been given by Malkowsky in . Also, Malkowsky gave general bounded linear operators on special BK spaces that are strongly summable to 0, summable and bounded with index equal to or greater than 1 . In , Kirisci gave well-known result related to some properties, dual spaces, and matrix transformations of the sequence space and introduced the matrix domain of space with arbitrary triangle matrix. The reflexivity of multiplication operators on some BK spaces with AK property has been studied in . Cyclicity of the adjoint of weighted composition operators on Hilbert function spaces, Fock spaces, and weighted Hardy spaces has been studied in [16–18]. In this section we want to study the cyclicity of the adjoint of a weighted composition operator acting on a space of formal power series with coefficients in a BK space with AK property. By the notations and we mean the point spectrum of and the open unit disc, respectively.
Theorem 1. Let be a BK space with AK. A complex number is a bounded point evaluation on if and only if and if and only if .
Proof. If is a bounded point evaluation it is clear that . Conversely, let and be a corresponding eigenvector in . For , we have Hence for all , we getand so Since we have Put . Then for all we have and this implies that for all polynomials . Since polynomials are dense in , is a bounded point evaluation and intact . Now it is clear that is a bounded point evaluation if and only if .
Theorem 2. Let be a BK space with AK and AD such that each point of is a bounded point evaluation on . Then a polynomial is cyclic for if and only if vanishes at no point in .
Proof. Let be such that for Fix and consider satisfying for all integers . Since , there exists such that for all Note thatfor all integers Since , we get and so for all But and ; hence for all and so Thus by the Hahn Banach Theorem, is cyclic for and so is a cyclic vector for The converse case is clear.
Theorem 3. Suppose that is a BK space with AK and AD, , is an analytic function on satisfying , and . Also, let be bounded and there exists satisfying for all , and assume that the set has a limit point in . Then is a cyclic vector for the operator acting on .
Proof. Let the map be given by . We prove that is continuous. For this we use the closed graph theorem. Suppose converges to in and converges to in . Then, for each in , Thus is convergent in . Now by the continuity of point evaluations converges pointwise to on . So is analytic and agrees with on . Hence and . Therefore, is continuouse and there is a constant such that for all in . But for all in . Thus for all Since and we will use instead of . Let ; then since . So since . On the other hand, note that, for all in , , where . Now we get , which implies that and so are bounded. Now, put . To complete the proof we show that if for all then should be the zero constant function. For this note that By the assumptions, clearly we get for all . Since has limit point in , it should be . Thus, is a cyclic vector for the operator acting on . This completes the proof.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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