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Full-derivable points of \(\mathcal {J}\)-subspace lattice algebras. (English) Zbl 1309.47041

Summary: Let \(\mathcal{L}\) be a \(\mathcal{J}\)-subspace lattice on a complex Banach space \(X\) and \(\mathrm {Alg}\,{\mathcal L}\) the associated \(\mathcal{J}\)-subspace lattice algebra. We say that an operator \(Z\in \mathrm {Alg}\,\mathcal L\) is a full-derivable point of \(\mathrm {Alg}\,{\mathcal L}\) if every linear map \(\delta\) from \(\mathrm {Alg}\,{\mathcal L}\) into itself derivable at \(Z\) (i.e., \(\delta(A)B+A\delta(B)=\delta(Z)\) for any \(A,B \in \mathrm {Alg}\,{\mathcal L}\) with \(AB=Z\)) is a derivation and is a full-generalized-derivable point of \(\mathrm {Alg}\,{\mathcal L}\) if every linear map \(\delta\) from \(\mathrm {Alg}\,{\mathcal L}\) into itself generalized derivable at \(Z\) (i.e., \(\delta(A)B+A\delta(B)-A\delta(I)B=\delta(Z)\) for any \(A,B \in \mathrm {Alg}\,{\mathcal L}\) with \(AB=Z\)) is a generalized derivation. In this paper, we prove that if \(Z\in\mathrm {Alg}\,{\mathcal L}\) is an injective operator or an operator with dense range, then \(Z\) is a full-derivable point as well as a full-generalized-derivable point of \(\mathrm {Alg}\,{\mathcal L}\).

MSC:

47B47 Commutators, derivations, elementary operators, etc.
47L35 Nest algebras, CSL algebras
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References:

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