Separation of two (possibly unbounded) components of the spectrum of a linear operator

Let X be a complex Banach space and A: D(A)→X a densely defined closed linear operator whose resolvent set contains the real line and for which {norm of matrix}λ(λ-A)-1{norm of matrix} is bounded on R. We give a necessary and sufficient condition, in terms of the complex powers of A and -A, for the existence of a decomposition X=X+⊕X-, where X± are closed subspaces, invariant for A, the spectra of the reduced operators A± are {λ∈σ(A);Imλ>0} and {λ∈σ(A);Imλ<0} respectively, and {norm of matrix}λ(λ-A±)-1{norm of matrix} is bounded for Imλ{less-than or greater-than}0. Finally we give an example of an operator in an Lp-type space for which the decomposition exists if 1<+∞ and does not exist if p=1. © 1989 Birkhäuser Verlag.