On Weyl’s theorem and its converse

Abstract

In this paper we study Weyl’s theorem and von Neumann’s converse of Weyl’s theorem for the classes of all operators of the form T f − 1 T g T_f^{ - 1}{T_g} and of the form T g T f − 1 {T_g}T_f^{ - 1} , where T g {T_g} and T f {T_f} are Toeplilz operators such that T f {T_f} is invertible; and we can prove that Weyl’s theorem holds for T f − 1 T g T_f^{ - 1}{T_g} and for T g T f − 1 {T_g}T_f^{ - 1} .