Maps Preserving Operator Pairs Whose Pencils And Jordan Products Are Partial Isometries

Partial isometries play an extremely important role in the theory of the polar decomposition as an important class of operators in the theory of the operators. Projections are a special class of partial isometries. They have the advantages of simple structure of spectrum, simple and convenient structure of the characterization of their properties. Therefore, scholars have made extensive research on projections both at home and abroad. And there have already been very deep research results with projections and their related properties as isomorphic invariants of the operator algebras. Partial isometries are unitary operators of two projective spaces in essence, and have very important algebra and geometry properties. Taking their algebra or geometry properties as isomorphic invariants caught scholars both at home and abroad attention. Therefore, this paper takes the pencil of operator as the research object and focuses on the characterization of maps preserving partial isometries of pencils of operators, and researches the characterization of maps preserving the nonzero partial isometries of Jordan product of two operators on the self-adjoint operator spaces at the same time. We have the following results in this paper:Firstly, we study the characterization of the map on B(H) preserving partial isometries of pencils of operators, that is, A-?B ? PI(H)(?)?(A)-??(B) ? PI(H), (?)A, B ? B(H),? G C. First of all, we put forward a sufficient and nec-essary condition to determine the partial ordering of two partial isometries. We verify the structure of the map is algebraic isomorphisms or anti-isomorphisms by the induction, basing on the sufficient and necessary condition, the algebra structure characteristics of the operator pencil and the characteristics of the map preserving the orthogonality and partial ordering of the partial isometries in both direction-s. When the dimension of H is less than 3, it is also true. All the above show that if we want to prove the two operator spaces are algebraic isomorphisms or anti-isomorphisms, we only need to preserve the algebraic structure of the partial isometries of pencils of operators between the two operator spaces.Secondly, the linear map preserving the nonzero partial isometries of Jordan product of two operators on the self-adjoint operator spaces is taken into consider-ation, that is, A(?)B ? PI*(H)??(A)(?)?(B)?PI*(H), (?)A, B?Bs(H). We prove that such a map preserves the unit or its negative, and combine its preserving partial ordering of the partial isometries, which make the description of the partial isometries translate into the description of the projections. When the dimension is not less than 3, the structure characteristics are obtained by Uhlhorn's theorem. Similar to the first part, the 2-dimension case is also true by using the Wigner's theorem. All the above show that if we want to prove the two self-adjoint operator spaces are algebraic isomorphisms or anti-isomorphisms, we only need to preserve the algebraic structure of the nonzero partial isometries of Jordan product of two operators between the two self-adjoint operator spaces.