| Preserver problem concerns the characterization of maps on operator algebras which leave certain functions,subsets,relations,etc.,invariant.The purpose of studying preserver problems is to reveal the connection between the inherent properties of operator algebras and maps on itself,which makes one know and understand operator algebras more deeply.The dissertation is devoted to the investigation of some preservers of reflexive algebras,including linear maps preserving p-similarity on B(X),linear maps preserving asymptotic similarity on B(X),linear maps preserving asymptotic equivalence on B(X),similarity Jordan multiplicative maps on B(X),linear maps on J-subspace lattice algebras which preserve similarity or equivalence.It consists of five chapters.In the first chapter,we introduce the background,review the developments and achievements until now.At the same time,we give preliminaries and the main results of this dissertation.In Chapter 2,we focus on some linear preservers on B(X),the Banach algebra of all bounded linear operators on Banach space X.We first investigate linear bijectionsΦ:B(X)→B(X)such that Φ(A)and Φ(B)are similar whenever A and B are psimilar.Here X is a complex Banach space with dimension at least 3.This result can be used to characterize Lie isomorphisms and Jordan isomorphisms.We also characterize linear bijections Φ:B(X)→B(X)such that Φ(A)and Φ(B)are asymptotically similar whenever A and B are similar.Here X is an infinite-dimensional complex Banach space.From this,we get that if H is an infinite-dimensional complex Hilbert space,then Φ:B(H)→B(H)is asymptotic similarity preserving if and only if it is similarity preserving.Finally,we study linear bijections Φ:B(X)→B(X)such that Φ(A)and Φ(B)are asymptotically equivalent whenever A and B are equivalent.Here X is a complex Banach space with dimension at least 2,From this,we get that if X is a complex Banach space with dimension at least 2,then a linear bijection Φ:B(X)→B(X)preserves asymptotic equivalence if and only if it preserves equivalence.In Chapter 3,applying the characterization of maps preserving orthogonality in both directions on the set of all idempotent operators in B(X),we study bijectionsΦ:B(X)→B(X)satisfying that Φ(AB+BA)and Φ(A)Φ(B)+Φ(B)Φ(A)are similar for all A,B∈B(X).Here X is a complex Banach space with dimension at least 3.In Chapter 4,we describe the structure of linear surjective maps which preserve any one of similarity and equivalence in both directions on reflexive algebras with Jsubspace lattices.The results can apply to atomic Boolean subspace lattice algebras and pentagon subspace lattice algebras,respectively.In Chapter 5,we summarize the whole text,and put forward some questions remaining unsolved. |
