The Bishop-Phelps-Bollobás properties in complex Hilbert spaces

Abstract

In this paper we consider a stronger property than the Bishop-Phelps-Bollobás property for various classes of operators on a complex Hilbert space. The Bishop-Phelps-Bollobás {\it point} property for some class $\mathcal{A}\subset \mathcal{L}(H)$ says that if one starts with a norm one operator $T$ belonging to $\mathcal{A}$, which almost attains its norm at some norm one vector $x_0$, then there is a new operator $S$, belonging to the same class $\mathcal{A}$, which is close to $T$ and attains its norm at the same vector $x_0$. We study it for classical operators on a complex Hilbert spaces such as self-adjoint, anti-symmetric, unitary, compact, normal, and Schatten-von Neumann operators. We also solve analogous problems by replacing the norm of an operator by its numerical radius.

14 pages