Property (b) and property (w) for operator matrices

Property (w) holds for T∈B(X) precisely when σa(T)∖σea(T)=π00(T). By comparison property (b) holds for T precisely when σa(T)∖σea(T)=p00(T). In this paper we explore how property (w) and property (b) survive for 2×2 operator matrices on the Banach space through local spectral theory. Also, we consider property (w) for the 2×2 skew-diagonal operator matrix of the form (0AB0) on the Hilbert space.