Cyclicity of Special Operators on a BK with AK Space

k=0 ∈ F, regardless of whether or not the series converges for any value of z. If F endowed with the norm of F, then F and F are norm isomorphic. Let ?̂?z : F → ?̂? be defined by (?̂?zf) = ∑ ∞ k=0 fkz , so the corresponding shift operator M : F → ω is defined by (Mf)n = fn−1 if n ≥ 1 and 0 else. A BK space is a Banach sequence space with the property that convergence implies coordinatewise convergence. A BK space F containing φ is said to have AK if every sequence f =


Introduction
We write  for the set of all complex sequences  = (  ) ∞ =0 .Let  denote the set of all finite sequences.By  () , we denote the sequence with  ()  = 1 and  ()  = 0 whenever  ̸ = .For any sequence  = (  ) ∞ =0 , let  [] = ∑  =0    () be its -section.Given any subset  of , we write F for the set of all formal power series f with f() = ∑ ∞ =0     where  = (  ) ∞ =0 ∈ , regardless of whether or not the series converges for any value of .If F endowed with the norm of , then  and F are norm isomorphic.Let M : F → ω be defined by ( M f) = ∑ ∞ =0    +1 , so the corresponding shift operator  :  →  is defined by ()  =  −1 if  ≥ 1 and 0 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  = (  ) ∞ =0 ∈  has a unique representation  = ∑ ∞ =0    () ; that is,  = lim  → ∞  [] ; 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 ()( p) = p().A point  is said to be a bounded point evaluation on F if the functional () extends to a continuous linear functional on F. Finally, we consider the multiplication of formal power series ĥ = f ĝ given by ĥ where ℎ  = ∑  =0    − for all integers  ≥ 0. If f ∈ F and p is a polynomial, then to the vector p(  ) f the formal power series p() f() corresponds.
Let Ω be a complex domain and let F be a Banach space of formal power series with coefficients in a reflexive BK space with AK such that F ⊂ (Ω).It is convenient and helpful to introduce the notation ⟨,  * ⟩ to stand for  * (), for  ∈ F and  * ∈ ( F) * .We assume 1 ∈ F and the operators M and the functional () of evaluation at  ( ∈ Ω) are bounded on F.
A complex valued function  on Ω for which  f ∈ F for every f ∈ F is called a multiplier of F and the collection of all of these multipliers is denoted by M( F).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 {   :  = 0, 1, 2, . ..}.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.

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., [1]).For  = (  ) ∞  , a sequence with   > 0 for all , Malkowsky considered the space V() = { ∈  : ∑ ∞ =0 |  −  −1 |   < ∞} and studied its -dual and characterized some linear operators on V() [2].In [3] the reflexivity of Λ-summable sequences from a Banach space is investigated whenever Λ is a Banach perfect sequence space.Some BK spaces including the spaces  0 and  have been introduced in [4] and also their duals have been computed.In [5], 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 [6].In [7], 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  0 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 [10] 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 [11], 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 [12].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 [13].In [14], Kirisci gave well-known result related to some properties, dual spaces, and matrix transformations of the sequence space V and introduced the matrix domain of space V with arbitrary triangle matrix.The reflexivity of multiplication operators on some BK spaces with AK property has been studied in [15].Cyclicity of the adjoint of weighted composition operators on Hilbert function spaces, Fock spaces, and weighted Hardy spaces has been studied in [16][17][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 Hence for all , we get and so Then for all  we have ⟨( (+1) ) ∧ , ()⟩ =  +1 and this implies that ⟨ p, ()⟩ = p() for all polynomials .Since polynomials are dense in F,  is a bounded point evaluation and intact () = ().

2 International
If M is a bounded operator on F, the adjoint ( M ) * : ( F) * → ( F) * satisfies ( M ) * () = ().In general each multiplier  of F determines a multiplication operator M defined by M f =  f, f ∈ F. Also ( M ) * () = ()().It is well-known that each multiplier is a bounded analytic function on Ω. Indeed |()| ≤ ‖ M ‖ for each  in Ω.Let  be an analytic self-map of the open unit disk .A composition operator   maps an analytic function f ∈ F into (  f)() = f(()).If  ∈ M( F) and   is bounded, Journal of Analysis     is called a weighted composition operator on M( F).
the notations   ( M *  ) and  we mean the point spectrum of M *  and the open unit disc, respectively.Let  be a BK space with AK.A complex number  is a bounded point evaluation on F if and only if  ∈   ( M *  ) and if and only if {  } ∞ =0 ∈   .Proof.If  is a bounded point evaluation it is clear that  ∈   (( M ) * ).Conversely, let  ∈   ( M *  ) and   = {   } ∞ =0 ∈   be a corresponding eigenvector in ( F) * .For  ∈ , we have Now it is clear that  is a bounded point evaluation if and only if{  } ∞ =0 ∈   . ( −   ) =   ( +1 −     ) = ℎ () +1 −   ℎ () Theorem 2. Let  be a BK space with AK and AD such that each point of  is a bounded point evaluation on F. Then a polynomial p is cyclic for M if and only if p vanishes at no point in .Proof.Let p() = ( −  1 ) ⋅ ⋅ ⋅ ( −   ) be such that   ∉  for  = 1, . . ., .Fix  ∈ {1, . . ., } and consider   ∈ ( F) * satisfying   ( M )  ( −   ) = 0 for all integers  ≥ 0. Since  * =   , there exists ℎ () ∈   such that   f = ⟨ f, ĥ()⟩ for all  ∈ .Note that   ( M )