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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 \(\text{alg } {\mathcal N}\) 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 \(\text{alg }{\mathcal N}\) 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.

MSC:
47L30 Abstract operator algebras on Hilbert spaces
47B47 Commutators, derivations, elementary operators, etc.
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