Let A and B be positive operators on a Banach lattice such that the commutator C = AB − B A is also positive. We study the size of the spectrum of C.

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Abstract. A local version of the Kleinecke-Shirokov theorem is proved. The results easily extend to bounded linear derivations on Banach algebras. In case of algebraic elements, an improved bound on the nilindex of a commutator is obtained as a consequence.Mathematics subject classification (2000): 47B47, 47B48, 46H05.

Abstract. Let X be a complex Banach space and e ∈ X a nonzero vector. Then the set of all operators T ∈ L(X) with σT (e) = σ δ (T ), respectively rT (e) = r(T ), is residual. This is an analogy to the well known result for a fixed operator and variable vector. The results are then used to characterize linear mappings preserving the local spectrum (or local spectral radius) at a fixed vector e.1. Introduction. Let X be a complex Banach space and let L(X) be the Banach algebra of all bounded linear operators on X. For T ∈ L(X), let σ(T ), σ δ (T ), and σ T (x) be the spectrum, the surjectivity spectrum, and the local spectrum of T at x ∈ X, respectively. We denote by r(T ) = lim k→∞ T k 1/k the spectral radius of T and by r T (x) = lim sup k→∞ T k x 1/k the local spectral radius of T at x. It is well known that r(T ) = max{|λ| : λ ∈ σ(T )}. On the other hand, r T (x) ≥ max{|λ| : λ ∈ σ T (x)}, and equality holds, for a given T and every nonzero x, if T has SVEP (see [6, Proposition 3.3.13 ]).For T ∈ L(X) and x ∈ X, it is easily seen that σ T (x) ⊆ σ δ (T ) ⊆ σ(T ). Moreover, the set of x ∈ X that satisfy σ δ (T ) = σ T (x) and r T (e) = r(T ) is residual (i.e., its complement is of the first category; see [8], and also [4], [6, Proposition 1.3.2]). In this paper, we are concerned with the "dual" question of how large the set D(e) = {T ∈ L(X) : σ δ (T ) = σ T (e)} is. It turns out (see Theorem 2.5) that D(e) is also residual whenever e = 0. Consequently, the set {T ∈ L(X) : r T (e) = r(T )}, which contains D(e), is residual as well.These results are later used to characterize linear mappings which preserve the local spectrum (local spectral radius) at a given vector e = 0. We show (Theorem 3.3) that a continuous linear surjection φ : L(X) → L(X) satisfies σ φ(T ) (e) = σ T (e), for every T ∈ L(X), if and only if there exists an invertible operator A ∈ L(X) such that Ae = e and φ(T ) = AT A −1 . This gives a solution to [5, Problem, p. 177].

Abstract. In this paper we introduce simple multipliers, a special subclass of multipliers on a Banach module. We show that, from a local spectral point of view, these multipliers behave like multipliers on a commutative Banach algebra. Our definition of simple multipliers relies on the notion of point multipliers. These multipliers were studied earlier. However our approach gives new insight into this topic and therefore could be of some interest by itself.2000 Mathematics Subject Classification. Primary 46H25; Secondary 47B40. The goal of the present paper is to discuss the decomposability of multipliers between Banach modules. Since, for a general multiplier, we cannot say very muchfor instance, if A = ‫,ރ‬ then B A (X) = B(X) -it is quite clear that we have to confine ourselves to a special subclass of multipliers. In Section 4 we introduce simple multipliers on a Banach left module, which seem to be the suitable environment for our questions. Section 5 is devoted to the local spectral theory of simple multipliers. For instance, we extend to the class of simple multipliers on a left Banach module some results that are proven in [8], [16] and [23] only for algebras. Our definition of simple multipliers relies on the notion of point multipliers, which are studied in Sections 2 and 3. We believe that the content of these two sections are of some interest by itself. Introduction. A mapping T on a Banach algebraIt is assumed that the reader is familiar with the concept of Banach modules. We refer to [1, 18, 24], for details.