Abstract
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)(x(1), x(2), ..., x(n)) is an (n - 1)th commutator defined by n indeterminates x(1), x(2), ..., x(n). It is shown that L satisfies the rule L(p(n)(A(1), ...
Abstract
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)(x(1), x(2), ..., x(n)) is an (n - 1)th commutator defined by n indeterminates x(1), x(2), ..., x(n). It is shown that L satisfies the rule L(p(n)(A(1), A(2), ..., A(n))) = Sigma(n)(k=1) p(n)(A(1), ..., A(k-1), L(A(k)), A(k+1), ...,A(n)) for all A(1), A(2), ..., A(n) AlgN if and only if there exist a linear derivation D: AlgN -> B(X) and a linear mapping H: AlgN -> CI vanishing on each (n - 1)th commutator p(n) (A(1), A(2), ..., A(n)) for all A(1), A(2), ...,A(n) is an element of AlgN such that L(A) = D (A) + H (A) for all A is an element of AlgN. We also propose some related topics for future research.
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