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Reinov, Oleg Ivanovich

 Statistics Math-Net.Ru Total publications: 25 Scientific articles: 25 Presentations: 2

 Number of views: This page: 2376 Abstract pages: 3753 Full texts: 1448 References: 189
Associate professor
Doctor of physico-mathematical sciences (2003)
Speciality: 01.01.01 (Real analysis, complex analysis, and functional analysis)
Birth date: 31.03.1951
E-mail:
Keywords: Approximation properties, nuclear operators, summing operators, vector measures, eigenvalues, quasinormed linear spaces, tensor products, Radon-Nikodym properties.
UDC: 513.83, 513.88, 517, 517.5, 513.8
MSC: 46b28, 47b38, 47b10, 46b20

Subject:

geometrical theory of linear operators in Banach spaces, absolutely summing and nuclear operators, bases and approximation conditions in normed spaces, tensor products of operators, vector measures, geometry of locally convex spaces, spectral theory.

Biography

I introduced in consideration (and investigated in details different properties, both geometrical and analytical): the Radon–Nikodym operators, the Radon–Nikodym sets. For the Theory of Radon-Nikodym operators I was awarded by Young Mathematician Prize of Leningrad Math. Society (1979).

I introduced (after P. Saphar, but in more general forms) in consideration and investigated in details different approximation properties of order $p\ge0$ in normed spaces, which generalize the well known approximation properties of A. Grothendieck.

I obtained many negative answers to a series of the questions of a french mathematician P. Saphar, of a german mathematician A. Pietsch, of a polish mathematician A. Peczynski, — the questions connected with the mentioned above approximation properties of order $p$, in particular to the question on the existence of $p$-nuclear operator with non-$p$-nuclear second adjoin. Moreover lately I found such the operators in Banach spaces with bases.

I obtained a negative answer to the question of A. Grothendieck (posed in 1955) on the equivalence the approximation and the bounded approximation properties for weakly compact operators.

I constructed counterexamples to some assertions, published by A. Grothendieck in 1955. For example, I showed that there are a Banach space $Z$ and an operator $U$ from $Z^{**}$ to $Z$ such that $Z^{**}$ has a basis, $U$ is nuclear but its second adjoint $U^{**}$ is not nuclear; A.  Grothendieck asserted that under these conditions (on $Z$ and $U$) $U^{**}$ must be nuclear (this was formulated in that work of A. Grothendieck at 1955 without a proof). My counterexample was published in Comp. Rendue in 1987.

I solved completely the following problem of Pelczynski–Pietsch: for which Banach spaces the operators, conjugate to which is $p$-nuclear, is dually $p$-nuclear itself (in "Problems of Calculus", 23, Novosibirsk, 2001, p. 147–205).

I, in cooperation with a Swedish mathematician Sten Kaijser, gave the negative answers to some questions of A. Defant and K. Floret (Defant A. and Floret F., "Tensor norms and operator ideals", North-Holland, Amsterdam, London, New York, Tokyo. 1993). Namely, we have proved that their tensor norms $g_\infty,$ $w_1$ and $w_\infty$ are not totally accessible.

I got a negative answer to A. Pietsch question: is it true that $T$ is $s$-nuclear if the adjoint $T^*$ is $s$-nuclear ($s\le1$)?

A lot of new theorems of Lidskii type on traces are obtained by me. Some of them were topics of the lectures at International Conferences (2009–2013). Trace formula was obtained by me, e.g., for several classes of nuclear operator in factor spaces of subspaces of $L_p$-spaces.

In cooperation with my PhD student Qaisar Latif (ASSMS Lahore), there was constructed a small part of spectral theory of $(r,p)$-nuclear operator (and corresponding approximation properties were introduced and investigated). Results were announced at several International Conferences in 2010–2013. Also, we obtained a negative answer to a question (in Math. Nachr.) posed by two well known Indian mathematicians; the result of ours was given, during the World International Mathematical Congress (2010) as a talk.

Main publications:
1. O.I. Reinov, “Operatory tipa RN v banakhovykh prostranstvakh”, Doklady AN SSSR, 220:3 (1975), 528–531
2. O.I. Reinov, “Svoistva approksimatsii poryadka $p$ i suschestvovanie ne $p$-yadernykh operatorov s $p$-yadernymi vtorymi sopryazhennymi”, Doklady AN SSSR, 256:1 (1981), 43–47
3. O.I. Reinov, “Un contre-exemple a une conjecture de A. Grothendieck”, C. R. Acad. Sc. Paris., Serie I, 296 (1983), 597–599
4. J. Bourgain, O.I. Reinov, “On the approximation properties for the space $H^\infty$”, Math. Nachr., 122 (1985), 19–27
5. O. Reinov, Q. Latif, “Grothendieck-Lidskii theorem for subspaces of $L_p$-spaces”, Math. Nachr., 286:2-3 (2013), 279–282

http://www.mathnet.ru/eng/person17628
List of publications on Google Scholar
https://zbmath.org/authors/?q=ai:reinov.oleg-i
https://mathscinet.ams.org/mathscinet/MRAuthorID/146595

Publications in Math-Net.Ru
 2020 1. O. I. Reinov, “A Banach Lattice Having the Approximation Property, but not Having the Bounded Approximation Property”, Mat. Zametki, 108:2 (2020),  252–259      ; Math. Notes, 108:2 (2020), 243–249 2. O. I. Reinov, “On the Product of $s$-Nuclear Operators”, Mat. Zametki, 107:2 (2020),  311–316      ; Math. Notes, 107:2 (2020), 357–362 2017 3. O. I. Reinov, “On products of nuclear operators”, Funktsional. Anal. i Prilozhen., 51:4 (2017),  90–91    ; Funct. Anal. Appl., 51:4 (2017), 316–317 2000 4. O. I. Reinov, “Approximation properties $\mathrm{AP}_s$ and $p$-nuclear operators (the case where $0 Presentations in Math-Net.Ru  1 Î ôàêòîðèçàöèè îïåðàòîðîâ â áàíàõîâûõ ïðîñòðàíñòâàõ (ñ íåêîòîðûìè ïðèìåíåíèÿìè)O. I. Reinov Seminar on Theory of Functions of Real VariablesNovember 1, 2019 18:30 2 The symmetry of a spectrum of nuclear operators in subspaces of$L_p\$-spacesO. I. Reinov International conference on Function Spaces and Approximation Theory dedicated to the 110th anniversary of S. M. Nikol'skiiMay 28, 2015 17:30

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