Motivated by the concept of disjoint topological transitivity for supercyclicity, in this paper, we consider the concept of disjoint Furstenberg-semi-transitivity for operators that are a composition of an isometric isomorphism and a left multiplier on a normed algebra. Thus, we characterize disjoint \(\mathcal {F}\) -semi-transitive and disjoint supercyclic such operators on a large class of non-unital normed algebras. It turns out that generalized weighted bilateral shifts on the standard Hilbert C*-module are just a special case of our theory. Generalized weighted composition operators on the normed algebra of operator-valued continuous functions vanishing at infinity on a locally compact, non-compact Hausdorff space are another special case of our theory. Next, we characterize disjoint \(\mathcal {F}\) -semi-transitive and disjoint supercyclic weighted composition operators on a large class of weighted solid Banach function spaces and we apply our results to the case of translations on weighted Morrey spaces. We illustrate all the results in this paper with concrete examples.
