In this article, we make a comprehensive study of the properties of multiplication operators acting on the spaces of functions of bounded \(p\)-variation in Wiener’s sense, \(WBV_p[0,1]\). We characterize all functions \(u\in WBV_p[0,1]\) that define invertible, compact and Fredholm multiplication operators \(M_u\) on \(WBV_p[0,1]\). Also we characterize when \(M_u\) has finite range and has closed range on \(WBV_p[0,1]\).
