Weyl type theorem and hypercyclic property for bounded linear operators

Let H be an infinite dimensional separable complex Hilbert space and B(H) be the algebra of all bounded linear operators on H. T∈B(H)satisfies Browder’s theorem if σ(T)\σw(T)π00(T) or σw(T)=σb(T). If σ(T)\σw(T)=π00(T), Weyl’s theorem holds for T, where σ(T), σw(T), σb(T) denote the spectrum set, Weyl spectrum, and Browder spectrum respectively, and π00(T)=λ∈iso σ(T):0<dim N (T－λI)<∞. Using the newly defined spectrum, the sufficient and necessary conditions for operator functions satisfying Weyl type theorem were studied if T is a hypercyclic operator. In addition, the spectrum mapping theorem for some new spectrums was discussed.