保不定半正交性的可加映射

ADDITIVE MAPS PRESERVING INDEFINITE SEMI-ORTHOGONALITY

设H_i是实或复数域上无限维完备的不定度规空间,A_i是B(H_i)中由单位元I和一个理想生成的子代数,其中B(H_i)表示H_i上所有有界线性算子构成的代数,i=1,2.本文刻画了从A_1到A_2上双边保不定半正交性的可加满射Φ,即对任意T,S∈A_1,T^+S= 0■Φ(T)^+Φ(S)=0.主要结果表明,这样的Φ具有形式Φ(T)=UTV对任意的T∈A_1成立,这里U,V是有界线性或共轭线性可逆算子且U^+U=cI,c是非零实数.

Let H be a complete indefinite inner product space of infinite dimension over real or complex field. A characterization of indefinite semi-orthogonality preserving additive surjection in both directions on unital t-subalgebras of B（H） which contains all finite rank operators is obtained. We show that such maps have the form φ（T） = UTV,arbitary T, where U and V are invertible linear or conjugate linear bounded operators with UU^＊ = cI for some nonzero real constant c. As a corollary, semi-orthogonality preserving additive surjections in both directions on unital ＊-subalgebra which contains all finite rank operators over Hilbert space are classified.