LIE-TYPE DERIVATIONS OF NEST ALGEBRAS ON BANACH SPACES AND RELATED TOPICS

Let X be a Banach space over the complex field C and B(X) be the algebra of all bounded linear operators on X. Let N be a nontrivial nest on X, AlgN be the nest algebra associated with N, and L: AlgN →B(X) be a linear mapping. Suppose that p_n(x₁, x₂, ⋯ , x_n) is an (n - 1) th commutator defined by n indeterminates x₁, x₂, ⋯ , x_n. It is shown that L satisfies the rule (equation presented) for all A₁, A₂, ⋯ , A_nϵ AlgN if and only if there exist a linear derivation D: AlgN →B(X) and a linear mapping H: AlgN-I vanishing on each (n - 1) th commutator pn(A₁, A₂, ⋯ , An) for all A₁, A₂, ⋯ , An ϵ AlgN such that L(A) = D(A) + H(A) for all A ϵ AlgN. We also propose some related topics for future research.