| Let A be an associative algebras.For A,B∈A,define the Jordan product of A,B by AB=AB+BA.Suppose the map δ:A→A is linear.Callδa derivation if δ(AB)=δ(A)B+Aδ(B)for allA,B∈A;a generalized derivation if 5(AB)=δ(A)B+Aδ(B)-Aδ(I)B for all A,B∈A;a Jordan derivation ifδ(A(?)B)＝δ(A)(?)B+Aoδ(B)for all A,B∈A.An active topic in the study of(Jordan)derivations is to find conditions under which a map is a.derivation.Along this,in this thesis we obtain the following resultsTheorem A.Let H be a,Hilbert space with dimension>1,W E B(H).Suppose the linear δ:B(H)→B(H)satisfies 8(A(?)B)=δ(A)(?)B+A(?)δ(B)for all A,B∈B(H)with A(?)B=W.Then δ is a generalized derivation when W=0;δ is a derivation when W≠0.Theorem B.Let X be a Banach space with dimension>1 and the map δ:B(X)→B(X)is linear.Ifδ(A)(?)A-1＋A(?)δ(A-1)=2δ(I)for all invertible operators A∈B(X)then δ is a derivation.Theorem C.Let H be a Hilbert space with dimension>1,AlgL be a nest algebra on H,δ:AlgL→AlgL is linear.If δ(A)(?)A-1+(?)δ(A-1)=2δ(I)for all invertible operators A ∈ AlgL,then δ is a derivation. |
