Banach空间C_p中Birkhoof正交性的两个刻画

TWO CHARACTERIZATIONS OF BIRKHOFF ORTHOGONALITY IN BANACH SPACE C_p

研究了Banach空间Cp中两元素A和B在B irkhoof意义下正交的条件,利用算子A的极分解A=U|A|,证明:当1<p<∞时,A正交于B的充要条件是tr(|A|p-1U*B)=0;利用端点概念,证明:当1<p<∞时,A正交于B当且仅当存在Cq的单位球的一个端点F满足tr(FA)=‖A‖p且tr(FB)=0。特别,两个紧算子A正交于B的充分必要条件是存在一个单位向量x∈H满足‖Ax‖=‖A‖及〈Ax,Bx〉=0。

Let A,B be two Cp -chass operators on a separable complex Hibert space H. The paper discusses some conditions for A to be orthogonal to B in the sense of Birkhoff. In the casewhere 1 〈p 〈 ∞ ,it is proved that A is orthogonal to B if and only if tr（｜A ｜^P-1 U ＊ B） =0 in light of the polar decomposition A = U｜A｜ of A. For 1 〈p〈∞,by using extreme point,it is shown that A is orthogonal to B if and only if there exists an extreme point F of the unit ball of Cq such that tr（FA） = ‖ A ‖ p and tr（FB） =0. It is also proved obtain that for two compact operators A and B,A is orthogonal to B is and only if there exists a unit vector x in H such that ‖ Ax ‖ = ‖ A ‖ and 〈Ax,Bx〉 =0.