Abstract: We considered a variant of the Weyl’s theorem: the generalized Weyl’s theorem. By defining a new spectral set, we gave sufficient and necessary conditions for the operator T and its functional calculus to satisfy the generalized Weyl’s theorem by using the spectral set. We obtained a new discriminant for T and its functional calculus to satisfy the generalized Weyl’s theorem.

Key words: Weyl’s theorem, generalized Weyl’s theorem, B-Weyl’s spectrum