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 Funktsional. Anal. i Prilozhen.: Year: Volume: Issue: Page: Find

 Funktsional. Anal. i Prilozhen., 2005, Volume 39, Issue 2, Pages 61–64 (Mi faa40)

Brief communications

Interpolation of Intersections Generated by a Linear Functional

S. V. Astashkin

Samara State University

Abstract: Let $(X_0,X_1)$ be a Banach couple such that $X_0\cap X_1$ is dense in $X_0$ and $X_1$. By $(X_0,X_1)_{\theta,q}$, $0<\theta<1$, $1\le q<\infty$, we denote the spaces of the real interpolation method. Let $\psi$ be a nonzero linear functional defined on some linear space $M\subset X_0+X_1$ and such that $\psi\in(X_0\cap X_1)^*$, and let $N=\operatorname{Ker}\psi$. We examine conditions under which the natural formula
$$(X_0\cap N,X_1\cap N)_{\theta,q}=(X_0,X_1)_{\theta,q}\cap N$$
is valid. In particular, the results obtained here imply those due to Ivanov and Kalton on the comparison of the interpolation spaces $(X_0,X_1)_{\theta,q}$ and $(N_0,X_1)_{\theta,q}$, where $\psi\in X_0^*$ and $N_0=\operatorname{Ker}\psi$. By way of application, we consider a problem, posed by Krugljak, Maligranda, and Persson, on the interpolation of intersections generated by an integral functional defined on weighted $L_p$-spaces.

Keywords: Banach space, interpolation space, subspace, Banach couple, subcouple, $\mathcal{K}$-functional, real interpolation method, weighted space

DOI: https://doi.org/10.4213/faa40

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English version:
Functional Analysis and Its Applications, 2005, 39:2, 131–134

Bibliographic databases:

UDC: 517.982.27

Citation: S. V. Astashkin, “Interpolation of Intersections Generated by a Linear Functional”, Funktsional. Anal. i Prilozhen., 39:2 (2005), 61–64; Funct. Anal. Appl., 39:2 (2005), 131–134

Citation in format AMSBIB
\Bibitem{Ast05} \by S.~V.~Astashkin \paper Interpolation of Intersections Generated by a Linear Functional \jour Funktsional. Anal. i Prilozhen. \yr 2005 \vol 39 \issue 2 \pages 61--64 \mathnet{http://mi.mathnet.ru/faa40} \crossref{https://doi.org/10.4213/faa40} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2161516} \zmath{https://zbmath.org/?q=an:1137.46010} \transl \jour Funct. Anal. Appl. \yr 2005 \vol 39 \issue 2 \pages 131--134 \crossref{https://doi.org/10.1007/s10688-005-0025-5} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000231004800005} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-23744479562} 

• http://mi.mathnet.ru/eng/faa40
• https://doi.org/10.4213/faa40
• http://mi.mathnet.ru/eng/faa/v39/i2/p61

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This publication is cited in the following articles:
1. S. V. Astashkin, P. Sunehag, “The Real Interpolation Method on Couples of Intersections”, Funct. Anal. Appl., 40:3 (2006), 218–221
2. Astashkin, SV, “Real method of interpolation on subcouples of codimension one”, Studia Mathematica, 185:2 (2008), 151
3. V. G. Malinov, “Nepreryvnyi metod minimizatsii vtorogo poryadka s operatorom proektirovaniya v peremennoi metrike”, Zhurnal SVMO, 21:1 (2019), 34–47
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