Norm-attaining operators which satisfy a Bollobás type theorem
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Títol
Norm-attaining operators which satisfy a Bollobás type theoremData de publicació
2021-04Editor
SpringerISSN
1735-8787; 2662-2033Cita bibliogràfica
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/s43037-020-00113-7Tipus de document
info:eu-repo/semantics/articleVersió de l'editorial
https://link.springer.com/article/10.1007/s43037-020-00113-7Versió
info:eu-repo/semantics/sumittedVersionParaules clau / Matèries
Resum
In this paper, we are interested in studying the set A(parallel to center dot parallel to) (X, Y) of all norm-attaining operators T from X into Y satisfying the following: given epsilon > 0, there exists eta such that ... [+]
In this paper, we are interested in studying the set A(parallel to center dot parallel to) (X, Y) of all norm-attaining operators T from X into Y satisfying the following: given epsilon > 0, there exists eta such that if parallel to Tx parallel to > 1 - eta, then there is x(0) such that parallel to x(0) - x parallel to < epsilon 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 A(parallel to center dot parallel to) (c(0), K). Also, we prove that the analogous result holds neither for A(parallel to center dot parallel to) (l(1), K) nor A(parallel to center dot parallel to) (l(infinity), K). Under some assumptions, we show that the sphere of the compact operators belongs to A(parallel to center dot parallel to) (X, Y) and that this is no longer true when some of these hypotheses are dropped. The analogous set A(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 A(parallel to center dot parallel to) (X, X) and A(nu)(X) when X = c(0) or l(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 A(parallel to center dot parallel to) (X, X) but not to A(nu)(X) and vice-versa. Finally, we establish some techniques which allow us to connect both sets by using direct sums. [-]
Descripció
This is a pre-print of an article published in Banach Journal of Mathematical Analysis. The final authenticated version is available online at: https://doi.org/10.1007/s43037-020-00113-7
Publicat a
Banach Journal of Mathematical Analysis volume, 2021, vol. 15, no 2Entitat finançadora
Estonian Research Council | NRF | Ministerio de Ciencia, Innovación y Universidades | Fondo Europeo de Desarrollo Regional (FEDER)
Codi del projecte o subvenció
PRG877 | NRF-2018R1A4A1023590 | FPU17/02023 | MTM2017-83262-C2-1-P
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