S(H)上保*-同构的线性映射（英文）

Linear Maps Preserving *-Congruence on S(H)

本文讨论了S(H)上保*-同构的线性映射.主要结论为:令H是复的有限维希尔伯特空间.我们记B(H)为H上的所有有界线性算子构成的Banach代数、S(H)为H上的所有自伴算子构成的实线性子空间,则有:为S(H)到S(H)的双边保*-同构的有界线性满射当且仅当存在B(H)上的可逆元A使得对所有T∈S(H)有(T)=ATA*.

In this paper,linear maps preserving ＊-congruence are discussed. The results we obtained are as follows： Let H be a complex infinite dimensional Hilbert space. By B（ H）,S（ H） we denote the Banach algebra of all bounded linear operators on H and the real linear subspace of all self-adjoint operators on H. It is proved that a bounded linear surjection Φ： S（ H）→S（ H） preserving ＊-congruence in both directions if and only if there exists invertible element A ∈ B（ H） such that Φ（ T） = AT A＊for all T ∈ S（ H）.