几个算子不等式的最佳常数

On the Best Constants for Some Operator Inequalities

该文给出了单位圆上解析函数中φ(z),|φ(z)<1,对于 Hilbert空间 H上任一真压缩算子 A,算子φ(A)的范数估计,得出 sup b^2(φ)=2, sup b^2(φ)=1. b(φ)=sup(||φ(A)||(1-||A||~2)+||φ(A)||~2)^(1/2),b(φ)=sup{||φ(A)||(1-||A||~2)+inf ||φ(A)x||~2}^(1/2),这里A取遍H上一切真压缩算子,并且若φ(z)=sum from σ=0 to n(0/σ),|φ(z)|,|φ(z)|<1,则||φ(A)||+||φ(A)||≤2n,其中φ(z)=z^nφ(z^(-1)),且2是与n,φ(z)及A无关常数中的最小者.

Let △ be the unit disk,and H, (△) be the follwing setH, (△)={φ(z):φ(z) is analytic in △, |φ(z)|<1,and there exists a θ∈[0,2π]such that |φ(e^(iθ))|=1}Then all the proper contraction on H,we havesup b^2(φ)=2,sup b^2(φ)=1φ∈H_1(△) φ∈H_1(△)where b^2(φ)=sup(||φ(A)||(1--||A||~2)+||φ(A)||~2) b^2(φ)=sup(||φ(A)||(1--||A||~2)+inf||φ(A)x||~2)On the other hand,ifφ(z)=sum from k=0 to n(0/k) a_kz^k,|φ(z)|<1, then ||φ(A)||+||φ(A)||≤2nwhere φ(z)=z^nφ(z^(-1)), and 2 is the best constant for all the choices of n,φ(z) and A.