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 fully characterized by the c0(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 c0(X)- and ℓ∞(X)-multiplier convergent series.