| A Quantitative Version of the Bishop–Phelps Theorem for Operators in Hilbert Spaces |
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| 基金项目： | The first author is supported by Natural Science Foundation of China （Grant No. 11071201）; the second author is supported by Natural Science Foundation of China （Grant No. 11001231） |
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| 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 ε 1/2. Then for every bounded linear operator T : H → H and x0 ∈ H with ||T|| = 1 = ||x0|| such that ||Tx0|| 1 ε, there exist xε∈ H and a bounded linear operator S : H → H with||S|| = 1 = ||xε|| such that ||Sxε|| = 1, ||xε-x0|| ≤ (2ε)1/2 + 4(2ε)1/2, ||S-T|| ≤(2ε)1/2.
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| 关 键 词： | Norm attaining operator Hilbert space Bishop–Phelps theorem |
A quantitative version of the Bishop-Phelps theorem for operators in Hilbert spaces |
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| Li Xin Cheng,Yun Bai Dong. A quantitative version of the Bishop-Phelps theorem for operators in Hilbert spaces[J]. Acta Mathematica Sinica(English Series), 2012, 28(10): 2107-2114. DOI: 10.1007/s10114-012-0537-x |
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| Authors: | Li Xin Cheng Yun Bai Dong |
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| Affiliation: | 1. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, P. R. China
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| Abstract: | 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 < ? < 1/2. Then for every bounded linear operator T: H → H and x 0 ∈ H with ‖T‖ = 1 = ‖x 0‖ such that ‖Tx 0‖ > 1 ? g3, there exist x ? ∈ H and a bounded linear operator S: H → H with ‖S‖ = 1 = ‖x ? ‖ such that $$left| {Sx_varepsilon } right| = 1, left| {x_varepsilon - x_0 } right| leqslant sqrt {2varepsilon } + sqrt[4]{{2varepsilon }}, left| {S - T} right| leqslant sqrt {2varepsilon } .$$ |
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| Keywords: | Norm attaining operator Hilbert space Bishop-Phelps theorem |
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