All-derivable points of nest algebras on Banach spaces (Q2914867)

Let \(X\) be a real or complex Banach space. A set \(\mathcal{N}\) of closed subspaces of \(X\), which is totally ordered by inclusion, is called a nest on \(X\) if \(\mathcal{N}\) contains \(\{0\}\), \(X\) and the intersections and the closed linear spans of all its subsets. The nest algebra \(\text{Alg}\mathcal{N}\) associated with the nest \(\mathcal{N}\) is the algebra consisting of the bounded linear operators on \(X\) leaving all the elements of \(\mathcal{N}\) invariant.NEWLINENEWLINEAn additive map \(\delta: \text{Alg}\mathcal{N}\rightarrow \text{Alg}\mathcal{N}\) is said to be derivable at \(Z\in \text{Alg}\mathcal{N}\) if, for all \(A, B\in \text{Alg}\mathcal{N}\) with \(AB=Z\), NEWLINE\[NEWLINE \delta(A B)=\delta(A)B+A\delta(B), NEWLINE\]NEWLINE and is called an additive derivation if the above equality holds for all \(A, B\in \text{Alg}\mathcal{N}\). In the present work, the authors address the problem of finding when it is the case that an additive mapping derivable at a point is automatically an additive derivation. These are the so-called additively all-derivable points. The authors give sufficient conditions for a point \(Z\in \text{Alg}\mathcal{N}\) to be an all-derivable point. More precisely, if an additive mapping \(\delta: \text{Alg}\mathcal{N}\rightarrow \text{Alg}\mathcal{N}\) is derivable at \(Z\in \text{Alg}\mathcal{N}\), then \(Z\) is an additively all-derivable point whenever any of the two following conditions hold:NEWLINENEWLINE(i) There exists a bounded idempotent operator \(P\) such that \(P(X)\in \mathcal{N} \) is non-trivial and complemented, \(PZP=Z \), and \(Z_{|_{P(X)}}\) is injective or has dense range in \(P(X)\).NEWLINENEWLINE(ii) There exists a bounded idempotent operator \(P\) such that \(P(X)\in \mathcal{N} \) is non-trivial and complemented, \((I-P)Z(I-P)=Z \), and \(Z_{|_{(I-P)(X)}}\) is injective or has dense range in \((I-P)(X)\).NEWLINENEWLINEIt is also shown that additive mappings on infinite dimensional Banach spaces having an additively all-derivable point satisfying (i) or (ii) are inner derivations, and that this does not necessarily hold in the finite dimensional case. As the authors point out, the results presented have an immediate extension to Hilbert spaces: one has only to require that \(P\) be an orthogonal projection, since in this case the range of \(P\) is complemented.