Bilinear Maps Preserving Zero Products

Many preserving problems on operator algebra have been researched. These results not only enrich theories of algebra but also have application on quantum mechanics and control theory. Preserving zero products is a new research direction on this basis. Over the last decade,many scholars have done a lot of work in this area. Such as maps preserving zero product on simple algebras; additive maps preserving zero product on B(H); additive maps preserving Jordan zero-product on nest algebras; bilinear maps preserving zero product on Banach space, and so on.This paper's main research work contains the following three parts:(1) We studied three properties related to bilinear maps preserving zero products, and relations among them. There are many properties of bilinear maps preserving zero products that role of each property varies. Here we studied three properties of them in order to change bilinear maps problem to linear maps problem according to these properties. Bilinear maps problem becomes simpler because of many good results on linear maps.(2) We discussed bilinear maps preserving anti-zero products, and proposed bilinear maps preserving anti-zero products based on bilinear maps preserving zero products. The form is opposite to preserving zero products. We obtained some relevant properties and their relations from the results. We proved that it can be characterized by a linear map on some conditions.(3) At last we characterized generalized anti-derivation with preserving zero products. We gave the definition of generalized anti-derivation by the definition of anti-derivation, and studied the relations between them. On certain conditions we can make transformation between them. Finally we can characterize generalized anti-derivation by using preserving zero products.