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February 2021 Semi-inner products and operators which attain their norm
Paweł Wójcik, Tomasz Stypuła
Proc. Japan Acad. Ser. A Math. Sci. 97(2): 7-11 (February 2021). DOI: 10.3792/pjaa.97.002

Abstract

It has been proved by Koehler and Rosenthal [Studia Math. 36 (1970), 213–216] that an linear isometry $U\in\mathcal{L}(X)$ preserves some semi-inner-product. Recently, similar investigations have been carried out by Niemiec and Wójcik for continuous representations of amenable semigroups into $\mathcal{L}(X)$ (cf. [Studia Math. 252 (2020), 27–48]).

In this paper we generalize the result of Koehler and Rosenthal. Namely, we prove that if an operator $T\in \mathcal{L}(X)$ of norm one attains its norm then there is a semi-inner-product $[\cdot|\diamond]: X\times X\to \mathbf{F}$ that the operator $T$ preserves this semi-inner-product on the norm attaining set. More precisely, we show that the equality $[T(\cdot)|\mathit{Tx}]=[\cdot|x]$ holds for all $x\in M_{T}:=\{y\in S_{X}: \|\mathit{Ty}\|=1\}$.

Citation

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Paweł Wójcik. Tomasz Stypuła. "Semi-inner products and operators which attain their norm." Proc. Japan Acad. Ser. A Math. Sci. 97 (2) 7 - 11, February 2021. https://doi.org/10.3792/pjaa.97.002

Information

Published: February 2021
First available in Project Euclid: 27 January 2021

Digital Object Identifier: 10.3792/pjaa.97.002

Subjects:
Primary: 46B20 , 46C50
Secondary: 46B10 , 47A05

Keywords: Bounded operator , norm attainment , semi-inner product

Rights: Copyright © 2021 The Japan Academy

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Vol.97 • No. 2 • February 2021
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