Reinov, Oleg Ivanovich

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Reinov, Oleg Ivanovich
Associate professor
Doctor of physico-mathematical sciences (2003)
Speciality: 01.01.01 (Real analysis, complex analysis, and functional analysis)
Birth date: 31.03.1951
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


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.


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
List of publications on Google Scholar

Publications in Math-Net.Ru
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  mathnet  mathscinet  elib; Math. Notes, 108:2 (2020), 243–249  isi  scopus
2. O. I. Reinov, “On the Product of $s$-Nuclear Operators”, Mat. Zametki, 107:2 (2020),  311–316  mathnet  mathscinet  elib; Math. Notes, 107:2 (2020), 357–362  isi  scopus
3. O. I. Reinov, “On products of nuclear operators”, Funktsional. Anal. i Prilozhen., 51:4 (2017),  90–91  mathnet  elib; Funct. Anal. Appl., 51:4 (2017), 316–317  isi  scopus
4. O. I. Reinov, “Approximation properties $\mathrm{AP}_s$ and $p$-nuclear operators (the case where $0<s<1$)”, Zap. Nauchn. Sem. POMI, 270 (2000),  277–291  mathnet  mathscinet  zmath; J. Math. Sci. (N. Y.), 115:2 (2003), 2243–2250
5. A. N. Podkorutov, O. I. Reinov, “On the Khinchin–Kahane inequality”, Algebra i Analiz, 10:1 (1998),  265–270  mathnet  mathscinet  zmath; St. Petersburg Math. J., 10:1 (1999), 211–215
6. O. I. Reinov, “Banach spaces without a local basis structure”, Mat. Zametki, 43:2 (1988),  220–228  mathnet  mathscinet  zmath; Math. Notes, 43:2 (1988), 124–129  isi
7. O. I. Reinov, “Metric space valued functions of the first Baire class and their applications”, Zap. Nauchn. Sem. LOMI, 135 (1984),  135–149  mathnet  mathscinet  zmath
8. O. I. Reinov, “How bad can a Banach space with the approximation property be?”, Mat. Zametki, 33:6 (1983),  833–846  mathnet  mathscinet  zmath; Math. Notes, 33:6 (1983), 427–434  isi
9. O. I. Reinov, “Banach spaces without approximation property”, Funktsional. Anal. i Prilozhen., 16:4 (1982),  84–85  mathnet  mathscinet  zmath; Funct. Anal. Appl., 16:4 (1982), 315–317  isi
10. O. I. Reinov, “Properties of $p$-order approximation and the existence of non-$p$-nuclear operators with $p$-nuclear second conjugate operators”, Dokl. Akad. Nauk SSSR, 256:1 (1981),  43–47  mathnet  mathscinet
11. O. I. Reinov, “Some remarks on the properties of Radon–Nikodým operators with applications to a problem of M. Talagrand”, Sibirsk. Mat. Zh., 22:1 (1981),  120–128  mathnet  mathscinet  zmath; Siberian Math. J., 22:1 (1981), 89–95  isi
12. O. I. Reinov, “Conditionally weakly compact and $(RN)^D$-operators”, Funktsional. Anal. i Prilozhen., 14:1 (1980),  83–84  mathnet  mathscinet  zmath; Funct. Anal. Appl., 14:1 (1980), 69–70
13. O. I. Reinov, “Some vector-lattice characterizations of operators of type $RN$”, Mat. Zametki, 27:4 (1980),  607–619  mathnet  mathscinet  zmath; Math. Notes, 27:4 (1980), 298–304  isi
14. O. I. Reinov, “Integral representations of linear operators that act from the space $L^1(\Omega,\Sigma,\mu)$”, Mat. Zametki, 27:2 (1980),  283–290  mathnet  mathscinet  zmath; Math. Notes, 27:2 (1980), 141–144  isi
15. O. I. Reinov, “A class of universally measurable maps”, Mat. Zametki, 26:6 (1979),  949–955  mathnet  mathscinet  zmath; Math. Notes, 26:6 (1979), 979–982  isi
16. O. I. Reinov, “On hereditarily dentable sets in Banach spaces”, Zap. Nauchn. Sem. LOMI, 92 (1979),  294–299  mathnet  mathscinet  zmath
17. E. D. Gluskin, S. V. Kislyakov, O. I. Reinov, “Tenson products of $p$-absolutely summing operators and right ($I_p$, $N_p$) multipliers”, Zap. Nauchn. Sem. LOMI, 92 (1979),  85–102  mathnet  mathscinet  zmath
18. O. I. Reinov, “Purely topological characteristics of operators of type $RN$”, Funktsional. Anal. i Prilozhen., 12:4 (1978),  89–90  mathnet  mathscinet  zmath; Funct. Anal. Appl., 12:4 (1978), 317–319
19. O. I. Reinov, “$RN$-sets in Banach spaces”, Funktsional. Anal. i Prilozhen., 12:1 (1978),  80–81  mathnet  mathscinet  zmath; Funct. Anal. Appl., 12:1 (1978), 63–64
20. O. I. Reinov, “Certain classes of continuous linear operations”, Mat. Zametki, 23:2 (1978),  285–296  mathnet  mathscinet  zmath; Math. Notes, 23:2 (1978), 154–159
21. O. I. Reinov, “Operators of $RN$ type in Banach spaces”, Sibirsk. Mat. Zh., 19:4 (1978),  857–865  mathnet  mathscinet  zmath; Siberian Math. J., 19:4 (1978), 606–612  isi
22. O. I. Reinov, “Geometric characterization of $RN$-operators”, Mat. Zametki, 22:2 (1977),  189–202  mathnet  mathscinet  zmath; Math. Notes, 22:2 (1977), 597–604
23. O. I. Reinov, “Certain classes of sets in Banach spaces and a topological characterization of operators of type $RN$”, Zap. Nauchn. Sem. LOMI, 73 (1977),  224–228  mathnet  mathscinet  zmath; J. Soviet Math., 34:6 (1986), 2156–2159
24. O. I. Reinov, “Operators of type $RN$ in Banach spaces”, Dokl. Akad. Nauk SSSR, 220:3 (1975),  528–531  mathnet  mathscinet  zmath
25. O. I. Reinov, “The Radon–Nikodym property and integral representations of linear operators”, Funktsional. Anal. i Prilozhen., 9:4 (1975),  87–88  mathnet  mathscinet  zmath; Funct. Anal. Appl., 9:4 (1975), 354–355

Presentations in Math-Net.Ru
1. О факторизации операторов в банаховых пространствах (с некоторыми применениями)
O. I. Reinov
Seminar on Theory of Functions of Real Variables
November 1, 2019 18:30
2. The symmetry of a spectrum of nuclear operators in subspaces of $L_p$-spaces
O. I. Reinov
International conference on Function Spaces and Approximation Theory dedicated to the 110th anniversary of S. M. Nikol'skii
May 28, 2015 17:30

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