Journal of Mathematical Analysis ISSN: 2217-3412, URL: www.ilirias.com/jma Volume 10 Issue 2 (2019), Pages 1-11. ON CERTAIN VECTOR VALUED MULTIPLIER SPACES AND SERIES OF OPERATORS MAHMUT KARAKUS ¸ Abstract. By L(X, Y ), we denote the space of all continuous linear operators between the normed spaces X and Y . In [15], Swartz introduced the (bounded) multiplier space for the series ∑ T j as: M ∞ (  T j )= {x =(x j ) ∈ ℓ∞(X)|  j T j x j converges}, where (T j ) ⊆ L(X, Y ). Recently in [6], Altay and Kama defined the vec- tor valued multiplier space M ∞ C (T ) of Ces`aro convergence by using Ces`aro summability method as follow: M ∞ C (T )= {x =(x k ) ∈ ℓ∞(X)|  k T k x k is Ces`aro convergent}. In this paper, we introduce the vector valued multiplier spaces S Λ (T ) and S wΛ (T ) by means of Λ− convergence and a sequence of continuous linear operators and study a series of some properties of these spaces. 1. Introduction ω is the space of all complex- (or real-) valued sequences and any subspace X of ω is called as a sequence space. A K space is a locally convex sequence space (lcss) X containing φ on which coordinate functionals π k (x)= x k are continuous for every k ∈ N, the set of natural numbers. Here φ is the space of finitely non-zero sequences spanned by the set {δ k : k ∈ N}. δ k is also the sequence whose only non-zero term is 1 in the k th address for all k ∈ N. Let X and Y be subspaces of ω and A =(a nk ), (∀n, k ∈ N) be an infinite matrix whose entries are the elements of R or C, the space of real numbers or complex numbers, respectively. Then A : X → Y may be considered as a linear transforma- tion of sequence x =(x k ) ∈ X by the expression y = Ax = {(Ax) n } =(y n ) with y =(y n ) ∈ Y is called as matrix transformation, that is, y n = ∑ k a nk x k , ∀n ∈ N [8]. For simplicity in notation, here and what follows, the summation without lim- its runs from 1 to ∞. Furthermore, the sequence x is said to be A- summable to a ∈ C if Ax converges to a which is called the A- limit of x. For recent results 2000 Mathematics Subject Classification. 46B15, 40A05, 46B45. Key words and phrases. Multiplier convergent series; Λ−summability; weakly unconditionally Cauchy series. c 2019 Ilirias Research Institute, Prishtin¨ e, Kosov¨ e. Submitted November 12, 2018. Published March 9, 2019. Communicated by Mikail Et. 1