arXiv subjects
Publications and source records attributed to Oleg I. Reinov.
It is shown how the results in the theory of determinants and traces as well as in the theory of quasi-normed tensor products can be applied for getting new theorems on distribution of eigenvalues of nuclear operators in Banach spaces and on coincidence of spectral and nuclear traces such operators. As examples, it is considered the new classes of operators -- generalized nuclear operators of Lorentz--Lapresté $N_{(r,s),p}.$
We study the possibilities of factorizations of products of nuclear operators of different types through the Schatten-von Neumann operators in Hilbert spaces with giving some applications to eigenvalues problems.
We answer in negative to the following question of Boris Mitjagin: Is it true that a product of two nuclear operators in Banach spaces can be factored through a trace class operator in a Hilbert space?
It was shown by M. I. Zelikin (2007) that the spectrum of a nuclear operator in a separable Hilbert space is central-symmetric iff the spectral traces of all odd powers of the operator equal zero. The criterium can not be extended to the case of general Banach spaces: It follows from Grothendieck-Enflo results that there exists a nuclear operator $U$ in the space $l_1$ with the property that $\operatorname{trace}\, U=1$ and $U^2=0.$ B. Mityagin (2016) has generalized Zelikin's criterium to the case of compact operators (in Banach spaces) some of which powers are nuclear. We give sharp generalizations of Zelikin's theorem (to the cases of subspaces of quotients of $L_p$-spaces) and of Mityagin's result (for the case where the operators are not necessarily compact).
It is a translation of an old paper of mine. We describe the topology tau_p in the space Pi_p(Y,X), for which the closures of convex sets in tau_p and in *-weak topology of the space Pi_p(Y,X) are coincident. Thereafter, we investigate some properties of the space Pi_p, related to this new topology. 2010-remark: Occasionally, the topology is coincides with the lambda_p-topology from the paper "Compact operators which factor through subspaces of l_p", Math. Nachr. 281(2008), 412-423 by Deba Prasad Sinha and Anil Kumar Karn.
A. Szankowski's example is used to construct a Banach space similar to that of "An example of an asymptotically Hilbertian space which fails the approximation property", P.G. Casazza, C.L. Garc\'ıa, W.B. Johnson [math.FA/0006134 (math@arXiv.org)].
Among other things, it is shown that there exist Banach spaces $Z$ and $W$ such that $Z^{**}$ and $W$ have bases, and for every $p\in[1,2)$ there is an operator $T:W\to Z$ that is not $p$-nuclear but $T^{**}$ is $p$-nuclear.
We give conditions on a pair of Banach spaces $X$ and $Y,$ under which each operator from $X$ to $Y,$ whose second adjoint factors compactly through the space $l^p,$ $1\le p\le+\infty$, itself compactly factors through $l^p.$ The conditions are as follows: either the space $X^*,$ or the space $Y^{***}$ possesses the Grothendieck approximation property. Leaving the corresponding question for parameters $p>1, p\neq 2,$ still open, we show that for $p=1$ the conditions are essential.
If $p\in [1,+\infty]$ and $T$ is a linear operator with $p$-nuclear adjoint from a Banach space $ X$ to a Banach space $Y$ then if one of the spaces $X^*$ or $Y^{***}$ has the approximation property, then $T$ belongs to the ideal $N^p$ of operators which can be factored through diagonal oparators $l_{p'}\to l_1.$ On the other hand, there is a Banach space $W$ such that $W^{**}$ has a basis and such that for each $p\in [1,+\infty], p\neq 2,$ there exists an operator $T: W^{**}\to W$ with $p$-nuclear adjoint that is not in the ideal $N^p,$ as an operator from $W^{**}$ to $ W.$