Quantitative version of the Bishop-Phelps-Bollob\'as theorem for operators with values in a space with the property β

The Bishop-Phelps-Bollob\'as property for operators deals with simultaneous approximation of an operator $T$ and a vector $x$ at which $T: X\rightarrow Y$ nearly attains its norm by an operator $F$ and a vector $z$, respectively, such that $F$ attains its norm at $z$. We study the possible estimates from above and from below for parameters that measure the rate of approximation in the Bishop-Phelps-Bollob\'as property for operators for the case of $Y$ having the property $\beta$ of Lindenstrauss.