Let H be an infinite dimensional separable complex Hilbert space and B(H) the algebra of all bounded linear operators on H.An operator T∈B(H) is said to satisfy Weyl′s theorem if σ(T)\\σw(T)=π00(T), where σ(T) and σw(T) denote the spectrum and Weyl spectrum of operator T respectively, and π00(T) denotes the set of all isolated eigenvalues of finite multiplicity.In this note,it is given that the definition of the Weyl-Kato decomposition of a bounded linear operator on Hilbert space. Using the new spectrum defined by the definition of the Weyl-Kato decomposition,it is established that sufficient and necessary conditions for Weyl′s theorem for the functions of operators.

Weyl-Kato decomposition; Weyl′s theorem; compact perturbations