Derivation On Nest Algebras Or Triangular Rings

The study of derivations, which are important both in theory and application, is an active topic in operator algebras and operator theory. In recent years, many mathematicians are intrested in idscussing the condition for linear (additive) maps to become derivation. For example, the study of the maps derivable at some point. Let R be a ring, an additive mapδfrom R into itself is called a derivation ifδ(AB)-δ(A)B+Aδ(B) for all A,B∈R. Let Z∈R. Ifδ(AB)=δ(A)B+Aδ(R) for any A,B∈R with AB= Z,we say thatδis derivable at Z. If every additive map from R into itself that is derivable at Z is a derivation, then Z is called an all-derivable point of R. This thesis focuses on the question what kind of points are all-derivable points of nest algebras or triangular rings. Some new type of all-derivable points are found, and thus several new characeterizations of derivations are obtained. The following are the main results obtained in this thesis:(1) let AlgN be a nest algebra on a complex Banach space X, and let N∈N. For any bounded idempotent operator P with range N, according to the space decompose of X determined by P. the operators in AlgN of the formΩ=(?) andΩ= (?) are all-derivable points of AlgN. whereΩ1 andΩ2 are injective operators or the operators with dense range.(2) Let A, B be unital rings with the property that 1/2I belongs to the ring and, for every T,there is some integer nT such that,nTI-Tis invertible.Then the elements of the formΩ=(0M00),Ω=(Ω1M00)andΩ=(0M0Ω2)are all-derivable points of the triangular ring U=Tri(A,B,M),whereΩ1 andΩ2 are invertible in A and B respectively,0≠M∈M.(3)Let A,B be unital rings with the property that 1/2 I,belongs to the ring and,for every T.there is some integer nT such that nTI-T is invertible.Then all invertible elements are all-derivable points of the triangular ring U=Tri(A,B,M).