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Trudy Inst. Mat. i Mekh. UrO RAN, 2019, Volume 25, Number 4, Pages 5–14 (Mi timm1665)  

On the conjugacy of the space of multipliers

V. V. Arestovab

a Ural Federal University named after the First President of Russia B. N. Yeltsin, Ekaterinburg
b Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg

Abstract: A. Figà Talamanca proved (1965) that the space $M_r=M_r(G)$ of bounded linear operators in the space $L_r$, $1\le r\le\infty$, on a locally compact group $G$ that are translation invariant (more exactly, invariant under the group operation) is the conjugate space for a space $A_r=A_r(G)$, which he described constructively. In the present paper, for the space $M_r=M_r(\mathbb{R}^m)$ of multipliers of the Lebesgue space $L_r(\mathbb {R}^m)$, $1\le r<\infty$, we present a Banach function space $F_r=F_r(\mathbb{R}^m)$ with two properties. The space $M_r$ is conjugate to $F_r$: $F^*_r=M_r$; actually, it is proved that $F_r$ coincides with $A_r=A_r(\mathbb{R}^m)$. The space $F_r$ is described in different terms as compared to $A_r$. This space appeared and has been used by the author since 1975 in the studies of Stechkin's problem on the best approximation of differentiation operators by bounded linear operators in the spaces $L_\gamma(\mathbb{R}^m)$, $1\le\gamma\le\infty$.

Keywords: predual space for the space of multipliers.

Funding Agency Grant Number
Russian Foundation for Basic Research 18-01-00336
Ministry of Education and Science of the Russian Federation 02.A03.21.0006
This work was supported by the Russian Foundation for Basic Research (project no. 18-01-00336) and by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University).


DOI: https://doi.org/10.21538/0134-4889-2019-25-4-5-14

Full text: PDF file (212 kB)
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Bibliographic databases:

UDC: 517.518+517.982
MSC: 47B38, 54C35
Received: 15.09.2019
Revised: 14.10.2019
Accepted:18.10.2019

Citation: V. V. Arestov, “On the conjugacy of the space of multipliers”, Trudy Inst. Mat. i Mekh. UrO RAN, 25, no. 4, 2019, 5–14

Citation in format AMSBIB
\Bibitem{Are19}
\by V.~V.~Arestov
\paper On the conjugacy of the space of multipliers
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2019
\vol 25
\issue 4
\pages 5--14
\mathnet{http://mi.mathnet.ru/timm1665}
\crossref{https://doi.org/10.21538/0134-4889-2019-25-4-5-14}
\elib{https://elibrary.ru/item.asp?id=41455516}


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