Weyl's theorem for the square of operator and perturbations

Let H be an infinite-dimensional separable complex Hilbert space and B(H) the algebra of all bounded linear operators on H. T is an element of B(H) satisfies Weyl's theorem if sigma(T)\sigma(w)(T) = pi(00)(T), where sigma(T) and sigma(w)(T) denote the spectrum and the Weyl spectrum of T, respectively, pi(00)(T) = {lambda is an element of iso sigma(T) : 0 < dim N(T - lambda I) < infinity}. T is an element of B(H) is said to have the stability of Weyl's theorem if T + K satisfies Weyl's theorem for all compact operator K is an element of B(H). In this paper, we characterize the operator T on H satisfying the stability of Weyl's theorem holds for T-2.