Left Derivable Maps at Non-Trivial Idempotents on Nest Algebras

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Let Alg be a nest algebra associated with the nest on a (real or complex) Banach space X. Suppose that there exists a non-trivial idempotent P Alg with range P (X), and δ : Alg Alg is a continuous linear mapping (generalized) left derivable at P, i.e. δ(ab) = (b) + (a) (δ(ab) = (b) + (a) baδ(I)) for any a, b Alg with ab = P, where I is the identity element of Alg . We show that δ is a (generalized) Jordan left derivation. Moreover, in a strongly operator topology we characterize continuous linear maps δ on some nest algebras Alg N with the property that δ(P ) = 2P δ(P ) or δ(P ) = 2P δ(P ) − P δ(I) for every idempotent P in Alg N .

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