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

Property (w) holds for T ∈ B ( X ) precisely when σ a ( T ) ∖ σ e a ( T ) = π 00 ( T ) . By comparison property (b) holds for T precisely when σ a ( T ) ∖ σ e a ( T ) = p 00 ( 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 ( 0 A B 0 ) on the Hilbert space.