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Norm-attaining operators which satisfy a Bollobás type theorem

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Abstract

In this paper, we are interested in studying the set \(\mathcal {A}_{\Vert \cdot \Vert }(X, Y)\) of all norm-attaining operators T from X into Y satisfying the following: given \(\varepsilon >0\), there exists \(\eta \) such that if \(\Vert Tx\Vert > 1 - \eta \), then there is \(x_0\) such that \(\Vert x_0 - x\Vert < \varepsilon \) and T itself attains its norm at \(x_0\). We show that every norm one functional on \(c_0\) which attains its norm belongs to \(\mathcal {A}_{\Vert \cdot \Vert }(c_0, \mathbb {K})\). Also, we prove that the analogous result holds neither for \(\mathcal {A}_{\Vert \cdot \Vert }(\ell _1, \mathbb {K})\) nor \(\mathcal {A}_{\Vert \cdot \Vert }(\ell _{\infty }, \mathbb {K})\). Under some assumptions, we show that the sphere of the compact operators belongs to \(\mathcal {A}_{\Vert \cdot \Vert }(X, Y)\) and that this is no longer true when some of these hypotheses are dropped. The analogous set \(\mathcal {A}_{{{\,\mathrm{nu}\,}}}(X)\) for numerical radius of an operator instead of its norm is also defined and studied. We present a complete characterization for the diagonal operators which belong to the sets \(\mathcal {A}_{\Vert \cdot \Vert }(X, X)\) and \(\mathcal {A}_{\text {nu}}(X)\) when \(X=c_0\) or \(\ell _{p}\). As a consequence, we get that the canonical projections \(P_N\) on these spaces belong to our sets. We give examples of operators on infinite dimensional Banach spaces which belong to \(\mathcal {A}_{\Vert \cdot \Vert }(X, X)\) but not to \(\mathcal {A}_{{{\,\mathrm{nu}\,}}}(X)\) and vice-versa. Finally, we establish some techniques which allow us to connect both sets by using direct sums.

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Acknowledgements

The authors would like to thank Manuel Maestre for suggesting the topic of the article and his helpful comments during his visit to POSTECH. They also would like to thank Miguel Martín and Abraham Rueda Zoca for fruitful conversations on the topic of the paper. They are also grateful to the anonymous referees for their useful suggestions. The first author was supported by the project OPVVV CAAS CZ.02.1.01/0.0/0.0/16_019/0000778 and by the Estonian Research Council grant PRG877. The second author was supported by NRF (NRF-2018R1A4A1023590). The third author was supported by the Spanish Ministerio de Ciencia, Innovación y Universidades, grant FPU17/02023 and by the MINECO and FEDER project MTM2017-83262-C2-1-P

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Correspondence to Mingu Jung.

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Communicated by Manuel Maestre.

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Dantas, S., Jung, M. & Roldán, Ó. Norm-attaining operators which satisfy a Bollobás type theorem. Banach J. Math. Anal. 15, 40 (2021). https://doi.org/10.1007/s43037-020-00113-7

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