Multiplication operators on the Banach algebra of bounded Φ-variation functions on compact subsets of ℂ

Abstract

Abstract Let A Φ ( K ) {{\mathbb{A}}}_{\Phi }\left({\bf{K}}) be the Banach algebra of bounded Φ \Phi -variation functions defined on a compact set K {\bf{K}} in the complex plane, h h a function defined on K {\bf{K}} , and M h {M}_{h} a multiplication operator induced by h h . In this article, we determine the conditions that h h must satisfy for M h {M}_{h} to be an operator that has closed range, finite rank or is compact. We also characterize the conditions that h h must satisfy for M h {M}_{h} to be a Fredholm operator.