The Bishop-Phelps-Bollobás property for Hermitian forms on Hilbert spaces (Q2922416)

Let \(H\) be a complex Hilbert space. In this paper, the authors exhibit several forms of the Bishop-Phelps-Bollobás theorem for continuous Hermitian forms. Let \(T\) be a self-adjoint operator of norm one, \(0< \epsilon <1\), \(x,y\) be unit vectors such that \(\langle T(x),y \rangle \geq 1-\frac{\epsilon^2}{4}\). Then there exists a self-adjoint operator \(R\), unit vectors \(x_1,y_1\) with \(\|R\| = 1= \langle R(x_1),y_1 \rangle\), \(\|T-R\| < \epsilon\), \(\max\{\|x-x_1\|, \|y-y_1\|\} < 4 \sqrt{\epsilon}\).