Abstract
We discuss range inclusion results for derivations on noncommutative Ba- nach algebras from the point of view of ring theory. 1. Results. Throughout, A always denotes a Banach algebra over the complex eld C. We denote by rad(A) the Jacobson radical of A and by r(x) the spectral radius of x2 A. Also, let Q(A) be the set of all quasinilpotent elements of A and let q-Inv(A) be the set of all quasi-regular elements in A. A linear mappingT : A! A is called spectrally bounded if there existsM 0 such that r(T (x)) Mr(x) for all x 2 A. In addition, if M = 0 (i.e., T (A) Q(A)), then T is called spectrally innitesimal . It is clear that rad(A) Q(A) q-Inv(A). Therefore, we have the following implications:
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