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Continuity and linearity of additive derivations of nest algebras on Banach spaces. (English) Zbl 0856.47028
Summary: 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 $\text{alg } {\cal N}$ 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 $\text{alg }{\cal N}$ such that the range of $p$ belongs to $\cal N$. It is also proved that every additive derivation of a nest algebra is automatically linear if the underlying normed space is infinite-dimensional.

47L30Abstract operator algebras on Hilbert spaces
47B47Commutators, derivations, elementary operators, etc.