Left Derivable Maps at Non-Trivial Idempotents on Nest Algebras

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

Abstract licence: CC BY-NC-ND

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