Vector-Valued Multiplier Spaces and Summing Operators: A Modulus Function Approach

In this paper, we introduce and systematically investigate novel classes of vector-valued multiplier spaces associated with operator-valued series, utilizing the concepts of ( f )-statistical and weak ( f )-statistical convergence. We begin by studying the topological properties of these newly defined spaces, establishing that their completeness is completely characterized by the ( c_0(X)- ) multiplier convergence of the underlying series. Building upon this structural foundation, we then explore the precise relationships between these ( f )-statistical spaces and classical statistical multiplier spaces, proving that they perfectly coincide under the assumption of a compatible modulus function. Furthermore, we define a natural summing operator acting on these spaces and conduct a detailed analysis of its mapping properties. By establishing necessary and sufficient conditions for the continuity and (weak) compactness of this summing operator, we obtain new characterizations for both ( c_0(X)- ) and ( ell_infty(X)- ) multiplier convergent series.