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arXiv · 1806.09361

The Bishop-Phelps-Bollob\'as properties in complex Hilbert spaces

Yun Sung Choi·Sheldon Dantas·Mingu Jung
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Abstract

In this paper we consider a stronger property than the Bishop-Phelps-Bollob\'{a}s property for various classes of operators on a complex Hilbert space. The Bishop-Phelps-Bollob\'as {\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.

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Yun Sung Choi, Sheldon Dantas, Mingu Jung. 2018-06-25. The Bishop-Phelps-Bollob\'as properties in complex Hilbert spaces. https://arxiv.org/abs/1806.09361

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