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.