Additive maps derivable or Jordan derivable at zero point on nest algebras |
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Authors: | Meiyan Jiao Jinchuan Hou |
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Institution: | a Department of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan 030006, PR China b Department of Mathematics, Shanxi University, Taiyuan 030006, PR China c Department of Mathematics, Taiyuan University of Technology, Taiyuan 030024, PR China |
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Abstract: | Let AlgN be a nest algebra associated with the nest N on a (real or complex) Banach space X. Assume that every N∈N is complemented whenever N-=N. Let δ:AlgN→AlgN be an additive map. It is shown that the following three conditions are equivalent: (1) δ is derivable at zero point, i.e., δ(AB)=δ(A)B+Aδ(B) whenever AB=0; (2) δ is Jordan derivable at zero point, i.e., δ(AB+BA)=δ(A)B+Aδ(B)+Bδ(A)+δ(B)A whenever AB+BA=0; (3) δ has the form δ(A)=τ(A)+cA for some additive derivation τ and some scalar c. It is also shown that δ is generalized derivable at zero point, i.e., δ(AB)=δ(A)B+Aδ(B)-Aδ(I)B whenever AB=0, if and only if δ is an additive generalized derivation. Finer characterizations of above maps are given for the case dimX=∞. |
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Keywords: | Primary: 47B49 47L35 |
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