Spaces of multiplier $\sigma$-convergent vector valued sequences and uniform $\sigma$-summability

This study focuses on the development of novel vector-valued sequence spaces whose elements are characterized by constructing (weakly) multiplier $\sigma$-convergent series. To achieve this, the concept of invariant means is rigorously examined and utilized as a foundational tool. These newly defined spaces are proven to possess the structure of Banach spaces when equipped with their natural sup norm, thus ensuring their completeness. In addition to establishing the Banach space properties, this study delves into the inclusion relationships between these new sequence spaces and classical multiplier spaces, specifically $BMC(B)$ and $CMC(B)$, where $B$ denotes an arbitrary Banach space. By employing the $\sigma$-convergence method, this study also culminates in a result analogous to the celebrated Hahn-Schur theorem, which traditionally establishes a connection between the weak convergence and the uniform convergence of unconditionally convergent series.