On a Brešar–šemrl Conjecture and Derivations of Banach Algebras

Locally quasi-nilpotent elementary operators

“…In order to gain information on the range of a linear mapping φ : A → A one can impose conditions on the size of the spectrum of the elements φ(x), x ∈ A. Various classes of linear mappings such as (inner) derivations or generalised (inner) derivations with small spectrum in the range, especially when the spectrum of every element in the range consists only of zero, have been studied by Aupetit, Brešar, Le Page, Pták, Šemrl, the present authors and many others; see, e.g., [3,9,5,6,18] and the references contained therein. Quite often such results are connected with commutativity criteria; see, e.g., [3,§2 in Chapter V].…”

Identifying Derivations Through the Spectra of Their Values

“…By studying (S) we follow the line of investigation of spectral properties of values of derivations. Let us mention some topics in this area: derivations and their products that have quasinilpotent values [9,18,22], spectrally bounded derivations [6], and derivations all of whose values have a finite spectrum [4,5,7]. A topic of a different kind, which, however, is closer to the problem considered here than it may seem at a first glance, is the study of derivations d and g such that the range of g(x) is contained in the range of d(x) for every x ∈ B.…”