A quantitative version of the Bishop-Phelps theorem for operators in Hilbert spaces

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 H and x(0) epsilon H with parallel to T parallel to = 1 = parallel to x (0)parallel to such that parallel to Tx (0)parallel to > 1 - epsilon, there exist x epsilon epsilon H and a bounded linear operator S : H -> H with parallel to S parallel to = 1 = parallel to x epsilon parallel to such that parallel to Sx epsilon parallel to = 1, parallel to x epsilon - x0 parallel to <= root 2 epsilon + 4 root 2 epsilon, parallel to S - T parallel to <= root 2 epsilonNatural Science Foundation of China [11071201, 11001231