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:

O.I. Reinov, “Operatory tipa RN v banakhovykh prostranstvakh”, Doklady AN SSSR, 220:3 (1975), 528–531

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

O.I. Reinov, “Un contre-exemple a une conjecture de A. Grothendieck”, C. R. Acad. Sc. Paris., Serie I, 296 (1983), 597–599

J. Bourgain, O.I. Reinov, “On the approximation properties for the space $H^\infty$”, Math. Nachr., 122 (1985), 19–27

O. Reinov, Q. Latif, “Grothendieck-Lidskii theorem for subspaces of $L_p$-spaces”, Math. Nachr., 286:2-3 (2013), 279–282

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 $0<s<1$)”, Zap. Nauchn. Sem. POMI, 270 (2000), 277–291; J. Math. Sci. (N. Y.), 115:2 (2003), 2243–2250

1998

5.

A. N. Podkorutov, O. I. Reinov, “On the Khinchin–Kahane inequality”, Algebra i Analiz, 10:1 (1998), 265–270; St. Petersburg Math. J., 10:1 (1999), 211–215

1988

6.

O. I. Reinov, “Banach spaces without a local basis structure”, Mat. Zametki, 43:2 (1988), 220–228; Math. Notes, 43:2 (1988), 124–129

1984

7.

O. I. Reinov, “Metric space valued functions of the first Baire class and their applications”, Zap. Nauchn. Sem. LOMI, 135 (1984), 135–149

1983

8.

O. I. Reinov, “How bad can a Banach space with the approximation property be?”, Mat. Zametki, 33:6 (1983), 833–846; Math. Notes, 33:6 (1983), 427–434

1982

9.

O. I. Reinov, “Banach spaces without approximation property”, Funktsional. Anal. i Prilozhen., 16:4 (1982), 84–85; Funct. Anal. Appl., 16:4 (1982), 315–317

1981

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

1980

11.

O. I. Reinov, “Conditionally weakly compact and $(RN)^D$-operators”, Funktsional. Anal. i Prilozhen., 14:1 (1980), 83–84; Funct. Anal. Appl., 14:1 (1980), 69–70

12.

O. I. Reinov, “Some vector-lattice characterizations of operators of type $RN$”, Mat. Zametki, 27:4 (1980), 607–619; Math. Notes, 27:4 (1980), 298–304

13.

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; Math. Notes, 27:2 (1980), 141–144

1979

14.

O. I. Reinov, “A class of universally measurable maps”, Mat. Zametki, 26:6 (1979), 949–955; Math. Notes, 26:6 (1979), 979–982

15.

O. I. Reinov, “On hereditarily dentable sets in Banach spaces”, Zap. Nauchn. Sem. LOMI, 92 (1979), 294–299

16.

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

1978

17.

O. I. Reinov, “Purely topological characteristics of operators of type $RN$”, Funktsional. Anal. i Prilozhen., 12:4 (1978), 89–90; Funct. Anal. Appl., 12:4 (1978), 317–319

18.

O. I. Reinov, “$RN$-sets in Banach spaces”, Funktsional. Anal. i Prilozhen., 12:1 (1978), 80–81; Funct. Anal. Appl., 12:1 (1978), 63–64

19.

O. I. Reinov, “Certain classes of continuous linear operations”, Mat. Zametki, 23:2 (1978), 285–296; Math. Notes, 23:2 (1978), 154–159

1977

20.

O. I. Reinov, “Geometric characterization of $RN$-operators”, Mat. Zametki, 22:2 (1977), 189–202; Math. Notes, 22:2 (1977), 597–604

21.

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; J. Soviet Math., 34:6 (1986), 2156–2159

1975

22.

O. I. Reinov, “Operators of type $RN$ in Banach spaces”, Dokl. Akad. Nauk SSSR, 220:3 (1975), 528–531

23.

O. I. Reinov, “The Radon–Nikodym property and integral representations of linear operators”, Funktsional. Anal. i Prilozhen., 9:4 (1975), 87–88; Funct. Anal. Appl., 9:4 (1975), 354–355