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

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 − g3, there exist xɛ ∈ H and a bounded linear operator S: H → H with ‖S‖ = 1 = ‖xɛ‖ such that $$\left\| {Sx_\varepsilon } \right\| = 1, \left\| {x_\varepsilon - x_0 } \right\| \leqslant \sqrt {2\varepsilon } + \sqrt[4]{{2\varepsilon }}, \left\| {S - T} \right\| \leqslant \sqrt {2\varepsilon } .$$