The Closed Range Property for Banach Space Operators

Abstract

Abstract. Let T be a bounded operator on a complex Banach space X. Let V be an open subset of the complex plane. We give a condition sufficient for the mapping f (z) → (T − z)f (z) to have closed range in the Fréchet space H(V, X) of analytic Xvalued functions on V . Moreover, we show that there is a largest open set U for which the map f (z) 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 and [12]. That for each T ∈ B(X) there exists 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,

Essential Spectral Inclusions for Operators on Banach Spaces

“…Moreover, by [14,Proposition 3], S β (T ) ⊆ σ ap (T ) for arbitrary T ∈ L(X). Moreover, T is said to satisfy the closed range condition (CR) on U if, for each open set V ⊆ U, the operator T V has closed range in H(V, X) (see [13], [14], and [15]). Thus T possesses property (β) on U precisely when T has both (SVEP) and (CR) on U.…”

“…Clearly, S(T ) is a closed subset of σ(T ). The corresponding result for condition (CR) is less obvious, and it was shown only recently, in [15,Theorem 4], that there exists a largest open set on which T has (CR). If the complement of this set is denoted by S cr (T ), then clearly To introduce the dual counterpart of Bishop's property (β), let…”

Closed Range Property for Holomorphic Semi-Fredholm Functions

“…where T ∈ L(X) is a bounded operator on a Banach space, closed range theorems of the above type are known and have been applied in the local spectral theory of Banach-space operators (see [2,13,14]). For instance, for every operator satisfying the finiteness condition dim(X/T X) < ∞, one can show that the generalized range R ∞ (T ) = k≥1 T k X of T has a representation of the form…”