Continuity and linearity of additive derivations of nest algebras on Banach spaces.

*(English)*Zbl 0856.47028Summary: This paper discusses the problem concerning the continuity and linearity of additive derivations of nest algebras on normed spaces. It is proved that every linear derivation of a nest algebra $\text{alg}\phantom{\rule{4.pt}{0ex}}\mathcal{N}$ is continuous provided that one of the following conditions is satisfied:

(1) ${0}_{+}\supset 0$,

(2) ${X}_{-}\subset X$,

(3) there exists a nontrivial idempotent $p$ in $\text{alg}\phantom{\rule{4.pt}{0ex}}\mathcal{N}$ such that the range of $p$ belongs to $\mathcal{N}$.

It is also proved that every additive derivation of a nest algebra is automatically linear if the underlying normed space is infinite-dimensional.