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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 alg𝒩 is continuous provided that one of the following conditions is satisfied:

(1) 0 + 0,

(2) X - X,

(3) there exists a nontrivial idempotent p in alg𝒩 such that the range of p belongs to 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:
47L30Abstract operator algebras on Hilbert spaces
47B47Commutators, derivations, elementary operators, etc.