Maps On B(H) Preserving Asymptotic Unitary Similarity And The Projections Of Products Of Two Operators

Preserver problems on operator algebras represent one of the most active re-search topics in operator algebra and also have important value in both theory and application for the classification of operator algebra. The research of this thesis fo-cuses on linear maps preserving asymptotic unitary similarity on operator algebras and non-linear surjective maps preserving the non-zero projections of products of operators on operator algebras. Using the technique of operator blocks, more char-acters of discussed maps can be found. This article is divided into three chapters.In chapter 1, the signigication and background of this thesis selecting subject are introduced. In addition, we offer some necesary notations, definitions, and some conceptions and conclusions in the later chapters. Firstly, we give the signigication and background of this thesis selecting subject. Subsequently, we introduce the definitions of asymptotic unitary similarity, minimal asymptotic u-similarity, invari-ant subspaces, the unitary orbit, one-rank projections etc. Finally, we give some well-known propositions and theorems.In chapter 2, the characters of the linear surjective mapφon B(H) preserv-ing asymptotic unitary similarity in both directions are studied. Let H be an infinite-dimensional complex Hilbert space and let B(H) denote the algebra of all bounded linear opeators on H. By giving the definition of minimal asymptotic uni-tary similarity-invariant subspaces and applying the closure of the unitary orbit of a linear operator, it is proved that the mapφhas the following representations:φ(X)= cU*XU orφ(X)=cU*XtU((?)X∈B(H)), where c is a non-zero constant, U∈B(H) is a unitary operator, and Xt denotes the transpose of X relative to a fixed but arbitrary orthonormal basis of H.In chapter 3, we analyse the related characters of the projections of products of two operators at first. Then, maps preserving the non-zero projections of products of two operators are characterized and the structures of the mentioned maps are obtained at last. Let B(H) be the algebra of all bounded linear operators on a complex Hilbert space H with dim H≤2. Letφbe a surjective map on B(H) preserving non-zero projections of products of two operators in both directions. By using the technique of operator blocks, it is proved that the mapφhas the following structure:φ(A)=λU*AU for all A in B(H) for some non-zero constantλwithλ2=1, where U is a unitary or an anti-unitary operator on H. These show that non-zero projections of products of two operators are isometric invariants of B(H).