Assume that the unit spheres of Banach spaces X and Y are uniformly homeomorphic.Then we prove that all unit spheres of the Lebesgue–Bochner function spaces Lp(μ, X) and Lq(μ, Y)are mutually uniformly homeomorphic ...Assume that the unit spheres of Banach spaces X and Y are uniformly homeomorphic.Then we prove that all unit spheres of the Lebesgue–Bochner function spaces Lp(μ, X) and Lq(μ, Y)are mutually uniformly homeomorphic where 1 ≤ p, q < ∞. As its application, we show that if a Banach space X has Property H introduced by Kasparov and Yu, then the space Lp(μ, X), 1 ≤ p < ∞,also has Property H.展开更多
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. Th...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.展开更多
基金The first author is supported by National Natural Science Foundation of China(Grant No.11471271)the second author is supported by the Foundation of Hubei Provincial Department of Education(Grant No.Q20161602)
文摘Assume that the unit spheres of Banach spaces X and Y are uniformly homeomorphic.Then we prove that all unit spheres of the Lebesgue–Bochner function spaces Lp(μ, X) and Lq(μ, Y)are mutually uniformly homeomorphic where 1 ≤ p, q < ∞. As its application, we show that if a Banach space X has Property H introduced by Kasparov and Yu, then the space Lp(μ, X), 1 ≤ p < ∞,also has Property H.
基金supported by Natural Science Foundation of China (Grant No. 11071201)supported by Natural Science Foundation of China (Grant No. 11001231)
文摘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.
