A Quantitative Version of the Bishop–Phelps Theorem for Operators in Hilbert Spaces  

作　　者：Li Xin CHENG Yun Bai DONG 

机构地区：[1]School of Mathematical Sciences,Xiamen University

出　　处：《Acta Mathematica Sinica,English Series》2012年第10期2107-2114,共8页数学学报（英文版）

基　　金：supported by Natural Science Foundation of China (Grant No. 11071201);supported by Natural Science Foundation of China (Grant No. 11001231)

摘　　要：In this paper, with the help of spectral integral, we show a quantitative version of the Bishop-Phelps theorem for operators in complex Hilbert spaces. Precisely, let H be a complex Hilbert space and 0 〈 s 〈 1/2. Then for every bounded linear operator T ： H → H and x0 ∈ H with ｜｜T｜｜ = 1 = ｜｜xo｜｜ such that ｜｜Txo｜｜ 〉 1-6, there exist xε ∈ H and a bounded linear operator S ： H → H with ｜｜S｜｜ = 1 = ｜｜xε｜｜ such that ｜｜Sxε｜｜=1, ｜｜x-ε0｜｜≤√2ε＋4√2ε, ｜｜S-T｜｜≤√2ε.In this paper, with the help of spectral integral, we show a quantitative version of the Bishop-Phelps theorem for operators in complex Hilbert spaces. Precisely, let H be a complex Hilbert space and 0 〈 s 〈 1/2. Then for every bounded linear operator T ： H → H and x0 ∈ H with ｜｜T｜｜ = 1 = ｜｜xo｜｜ such that ｜｜Txo｜｜ 〉 1-6, there exist xε ∈ H and a bounded linear operator S ： H → H with ｜｜S｜｜ = 1 = ｜｜xε｜｜ such that ｜｜Sxε｜｜=1, ｜｜x-ε0｜｜≤√2ε＋4√2ε, ｜｜S-T｜｜≤√2ε.

关 键 词：Norm attaining operator Hilbert space Bishop-Phelps theorem 

分 类 号：O177.1[理学—数学] O174[理学—基础数学]