On locally linearly dependent operators and derivations

The first section of the paper deals with linear operators T-i : U --> V, i = 1,..., n, where U and V are vector spaces over an infinite field, such t hat for every u is an element of U, the vectors T-1 u,..., T-n u are linear ly dependent module a fixed finite dimensional subspace of V. In the second section, outer derivations of dense algebras of linear operators are discu ssed. The results of the first two sections of the paper are applied in the last section, where commuting pairs of continuous derivations d,g of a Ban ach algebra A such that (dg)(x) is quasi-nilpotent for every x is an elemen t of A are characterized.