On locally linearly dependent operators and derivations

The first section of the paper deals with linear operators $T_i:U\longrightarrow V$, $i = 1,\ldots,n$, where $U$ and $V$ are vector spaces over an infinite field, such that for every $u \in U$, the vectors $T_1 u,\ldots,T_n u$ are linearly dependent modulo a fixed finite dimensional subspace of $V$. In the second section, outer derivations of dense algebras of linear operators are discussed. 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 Banach algebra ${\cal A}$ such that $(dg)(x)$ is quasi-nilpotent for every $x \in {\cal A}$ are characterized.