Spectrum of interpolated operators

Let (X0,X1) be a compatible pair of Banach spaces and let T be an operator that acts boundedly on both X0 and X1. Let T[θ] (0 ≤ θ ≤ 1) be the corresponding operator on the complex interpolation space (X0, X1)[θ]. The aim of this paper is to study the spectral properties of T[θ]. We show that in general the set-valued function θ 7→ σ(T[θ]) is discontinuous even in inner points θ ∈ (0, 1) and show that each operator satisfies the local uniquenessof-resolvent condition of Ransford. Further we study connections with the real interpolation method. I. Complex interpolation Let X̄ = (X0, X1) be a compatible pair of Banach spaces, i.e., (X0, ‖ · ‖0) and (X1, ‖ · ‖1) are Banach spaces continuously embedded in a Hausdorff topological vector space. Then X̄∆ = X0 ∩ X1 and X̄Σ = X0 + X1 endowed with norms ‖x‖∆ = max{‖x‖0, ‖x‖1} and ‖x‖Σ = inf{‖a‖0 + ‖b‖1 : a ∈ X0, b ∈ X1, a+ b = x} are Banach spaces. Recall the construction of complex interpolation spaces X̄[θ] (0 ≤ θ ≤ 1) (see e.g. [C], [BL], [T]). Let G = {z ∈ C : 0 < Re z < 1}. Denote by F the set of all continuous functions f : Ḡ → X̄Σ that are analytic on G such that f(j + it) ∈ Xj (j = 0, 1, t ∈ R) and lim|t|→∞ ‖f(j + it)‖j = 0 (j = 0, 1). Then F with the norm ‖f‖F = max { max t∈R ‖f(it)‖0,max t∈R ‖f(1 + it)‖1 } becomes a Banach space. The intermediate spaces X̄[θ] are defined by X̄[θ] = {f(θ) : f ∈ F} with the norm ‖x‖[θ] = inf{‖f‖F : f(θ) = x}. Then X̄∆ ⊂ X̄[θ] ⊂ X̄Σ and X̄∆ is dense in X̄[θ] (0 ≤ θ ≤ 1). Further X̄[0] and X̄[1] are closed subspaces of X0 and X1, respectively. Clearly X̄[θ] can be identified with the quotient space F/Nθ where Nθ = {f ∈ F : f(θ) = 0}. Let T : X̄Σ → X̄Σ be a linear mapping such that TX0 ⊂ X0, TX1 ⊂ X1 and the restrictions T |Xj are bounded with respect to the norms on Xj (j = 0, 1) (we denote this situation by T : X̄ → X̄). Then T X̄[θ] ⊂ X̄[θ] for all θ; the restriction T |X̄[θ] is denoted by T[θ]. Received by the editors September 25, 1998 and, in revised form, May 14, 1999. 2000 Mathematics Subject Classification. Primary 46B70, 47A10.