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 Mat. Zametki, 2011, Volume 89, Issue 4, Pages 547–557 (Mi mz7699)

Universal Spaces of Subdifferentials of Sublinear Operators Ranging in the Cone of Bounded Lower Semicontinuous Functions

Institute of System Dynamics and Control Theory, Siberian Branch of the Russian Academy of Sciences

Abstract: We study Fréchet's problem of the universal space for the subdifferentials $\partial P$ of continuous sublinear operators $P\colon V\to BC(X)_{\sim}$ which are defined on separable Banach spaces $V$ and range in the cone $BC(X)_\sim$ of bounded lower semicontinuous functions on a normal topological space $X$. We prove that the space of linear compact operators $L^{\mathrm c}(\ell^2,C(\beta X))$ is universal in the topology of simple convergence. Here $\ell^2$ is a separable Hilbert space, and $\beta X$ is the Stone–Ĉech compactification of $X$. We show that the images of subdifferentials are also subdifferentials of sublinear operators.

Keywords: sublinear operator, subdifferential, topology of simple convergence, lower semicontinuous function, Fréchet problem for universal spaces, separable Banach space, continuous selection

DOI: https://doi.org/10.4213/mzm7699

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English version:
Mathematical Notes, 2011, 89:4, 519–527

Bibliographic databases:

UDC: 517.982+517.988
Revised: 18.11.2010

Citation: Yu. E. Linke, “Universal Spaces of Subdifferentials of Sublinear Operators Ranging in the Cone of Bounded Lower Semicontinuous Functions”, Mat. Zametki, 89:4 (2011), 547–557; Math. Notes, 89:4 (2011), 519–527

Citation in format AMSBIB
\Bibitem{Lin11} \by Yu.~E.~Linke \paper Universal Spaces of Subdifferentials of Sublinear Operators Ranging in the Cone of Bounded Lower Semicontinuous Functions \jour Mat. Zametki \yr 2011 \vol 89 \issue 4 \pages 547--557 \mathnet{http://mi.mathnet.ru/mz7699} \crossref{https://doi.org/10.4213/mzm7699} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2856746} \transl \jour Math. Notes \yr 2011 \vol 89 \issue 4 \pages 519--527 \crossref{https://doi.org/10.1134/S0001434611030230} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000290038700023} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-79955597728} 

• http://mi.mathnet.ru/eng/mz7699
• https://doi.org/10.4213/mzm7699
• http://mi.mathnet.ru/eng/mz/v89/i4/p547

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This publication is cited in the following articles:
1. I. V. Orlov, Z. I. Khalilova, “Compact subdifferentials in Banach spaces and their applications to variational functionals”, Journal of Mathematical Sciences, 211:4 (2015), 542–578
2. I. V. Orlov, “Introduction to sublinear analysis”, Journal of Mathematical Sciences, 218:4 (2016), 430–502
3. I. V. Orlov, S. I. Smirnova, “Invertibility of multivalued sublinear operators”, Eurasian Math. J., 6:4 (2015), 44–58
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