The Closed Range Property for Banach Space Operators

Let T be a bounded operator on a complex Banach space X. If V is an open subset of the complex plane, we give a condition sufficient for the mapping f(z) 7→ (T − z)f(z) to have closed range in the Fréchet space H(V, X) of analytic X-valued functions on V . Moreover, we show that there is a largest open set U for which the map f(z) 7→ (T − z)f(z) has closed range in H(V, X) for all V ⊆ U . Finally, we establish analogous results in the setting of the weak–∗ topology on H(V, X∗). Introduction. Let X be a complex Banach space and denote by B(X) the algebra of bounded linear operators on X. For T ∈ B(X), let σ(T ) denote the spectrum of T , and denote by Lat (T ) the collection of closed T -invariant subspaces of X. If M ∈ Lat (T ), we write the restriction of T to M as T |M . A basic notion in local spectral theory is that of decomposability. Given an open subset U of the complex plane C, T ∈ B(X) is said to be decomposable on U provided that for any open cover {V1, . . . , Vn} of C with C \ U ⊂ V1, there exists {X1, . . . , Xn} ⊂ Lat (T ) such that X = X1 + · · · + Xn and σ(T |Xk) ⊂ Vk for each k, 1 ≤ k ≤ n; see [2], [5], [8], [11], and [12]. The fact that there exists for each T ∈ B(X) a largest open set U on which T is decomposable was first shown by Nagy, [11]. An alternative characterization of decomposability may be given in terms of a property introduced by E. Bishop, [3]. For an open subset V of C, let H(V,X) denote the space of all analytic X-valued functions on V. Then H(V,X) is a Fréchet space with generating semi-norms given by pK(f) := sup {‖f(λ)‖ : λ ∈ K} , where K runs through the compact subsets of V. Every operator T ∈ B(X) induces a continuous linear mapping TV on H(V,X), defined by TV f(λ) := (T − λ)f(λ) for all f ∈ H(V, X) and λ ∈ V. An operator T is said to possess Bishop’s property (β) on an open set U ⊂ C if for each open subset V of U, the operator TV is injective with range ran TV closed in H(V, X); see [6, Prop. 1.2.6]. Clearly there exists a largest open set ρβ(T ) on which T has property (β). Fundamental work by Albrecht and Eschmeier established that an operator T ∈ B(X) has property (β) on U precisely when there exists an operator S ∈ B(Y ) such that S is decomposable on U , X ∈ Lat (S) and T = S|X , [2, Theorem 10]. Moreover, [2, Theorems 8 and 21], T is decomposable on U if and only if T and its adjoint T ∗ share property (β) on U . Thus Nagy’s largest open set on which T is decomposable is the set ρβ(T ) ∩ ρβ(T ∗). Part of this work has been prepared while the first author was a guest of the Mathematical Institute of the Czech Academy of Sciences. He would like to express his gratitude to the Institute and to his coauthor for his hospitality. The second author was supported by grant No. 201/06/0128 of GA ČR and by Institutional Research Plan AV 0Z 10190503. 1 Preprint, Institute of Mathematics, AS CR, Prague. 2007-12-21 I N T IT U TE of M ATH TICS A ca d em y o f Sc ie n ce s C ze ch R ep u b lic