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Timur Oikhberg

Publications and source records attributed to Timur Oikhberg.

28 records · Page 2Linked to original sources
arXiv2013

Domination of operators in the non-commutative setting

We consider majorization problems in the non-commutative setting. More specifically, suppose $E$ and $F$ are ordered normed spaces (not necessarily lattices), and $0 \leq T \leq S :E \to F$. If $S$ belongs to a certain ideal (for instance, the ideal of compact or Dunford-Pettis operators), does it follow that $T$ belongs to that ideal as well? We concentrate on the case when $E$ and $F$ are $C^*$-algebras, preduals of von Neumann algebras, or non-commutative function spaces. In particular, we show that, for $C^*$-algebras $\A$ and ${\mathcal{B}}$, the following are equivalent: (1) at least one of the two conditions holds: (i) $\A$ is scattered, (ii) ${\mathcal{B}}$ is compact; (2) if $0 \leq T \leq S : \A \to {\mathcal{B}}$, and $S$ is compact, then $T$ is compact.

math.OA↗
arXiv2012

Lebesgue type inequalities for quasi-greedy bases

We show that for quasi-greedy bases in real or complex Banach spaces the error of the thresholding greedy algorithm of order N is bounded by the best N- term error of approximation times a function of N which depends on the democracy functions and the quasi-greedy constant of the basis. If the basis is democratic this function is bounded by C logN. We show with two examples that this bound is attained for quasi-greedy democratic bases.

math.FA↗
arXiv2010

Rate of decay of s-numbers

For an operator $T \in B(X,Y)$, we denote by $a_m(T)$, $c_m(T)$, $d_m(T)$, and $t_m(T)$ its approximation, Gelfand, Kolmogorov, and absolute numbers. We show that, for any infinite dimensional Banach spaces $X$ and $Y$, and any sequence $α_m \searrow 0$, there exists $T \in B(X,Y)$ for which the inequality $$ 3 α_{\lceil m/6 \rceil} \geq a_m(T) \geq \max\{c_m(t), d_m(T)\} \geq \min\{c_m(t), d_m(T)\} \geq t_m(T) \geq α_m/9 $$ holds for every $m \in \N$. Similar results are obtained for other $s$-scales.

math.FA↗
arXiv2010

Subspace structure of some operator and Banach spaces

We construct a family of separable Hilbertian operator spaces, such that the relation of complete isomorphism between the subspaces of each member of this family is complete $\ks$. We also investigate some interesting properties of completely unconditional bases of the spaces from this family. In the Banach space setting, we construct a space for which the relation of isometry of subspaces is equivalent to equality of real numbers.

math.FA↗
arXiv2010

Some Results on Metric Trees

Using isometric embedding of metric trees into Banach spaces, this paper will investigate barycenters, type and cotype, and various measures of compactness of metric trees. A metric tree ($T$, $d$) is a metric space such that between any two of its points there is an unique arc that is isometric to an interval in $\mathbb{R}$. We begin our investigation by examining isometric embeddings of metric trees into Banach spaces. We then investigate the possible images $x_0=π((x_1+\ldots+x_n)/n)$, where $π$ is a contractive retraction from the ambient Banach space $X$ onto $T$ (such a $π$ always exists) in order to understand the "metric" barycenter of a family of points $ x_1, \ldots,x_n$ in a tree $T$. Further, we consider the metric properties of trees such as their type and cotype. We identify various measures of compactness of metric trees (their covering numbers, $ε$-entropy and Kolmogorov widths) and the connections between them. Additionally, we prove that the limit of the sequence of Kolmogorov widths of a metric tree is equal to its ball measure of non-compactness.

math.MG↗
arXiv2007

Rosenthal operator spaces

In 1969 Lindenstrauss and Rosenthal showed that if a Banach space is isomorphic to a complemented subspace of an L_p-space, then it is either a script L_p-space or isomorphic to a Hilbert space. This is the motivation of this paper where we study non--Hilbertian complemented operator subspaces of non commutative L_p-spaces and show that this class is much richer than in the commutative case. We investigate the local properties of some new classes of operator spaces for every $2<p< \infty$ which can be considered as operator space analogues of the Rosenthal sequence spaces from Banach space theory, constructed in 1970. Under the usual conditions on the defining sequence sigma we prove that most of these spaces are operator script L_p-spaces, not completely isomorphic to previously known such spaces. However it turns out that some column and row versions of our spaces are not operator script L_p-spaces and have a rather complicated local structure which implies that the Lindenstrauss--Rosenthal alternative does not carry over to the non-commutative case.

math.FA↗
arXiv2002

A theorem of Krein revisited

M. Krein proved in 1948 that if T is a continuous operator on a normed space leaving invariant an open cone, then its adjoint T* has an eigenvector. We present generalizations of this result as well as some applications to C*-algebras, operators on l_1, operators with invariant sets, contractions on Banach lattices, the Invariant Subspace Problem, and von Neumann algebras.

math.FA↗
arXiv1999

On certain extension properties for the space of compact operators

Let $Z$ be a fixed separable operator space, $X\subset Y$ general separable operator spaces, and $T:X\to Z$ a completely bounded map. $Z$ is said to have the Complete Separable Extension Property (CSEP) if every such map admits a completely bounded extension to $Y$; the Mixed Separable Extension Property (MSEP) if every such $T$ admits a bounded extension to $Y$. Finally, $Z$ is said to have the Complete Separable Complementation Property (CSCP) if $Z$ is locally reflexive and $T$ admits a completely bounded extension to $Y$ provided $Y$ is locally reflexive and $T$ is a complete surjective isomorphism. Let ${\bf K}$ denote the space of compact operators on separable Hilbert space and ${\bf K}_0$ the $c_0$ sum of ${\Cal M}_n$'s (the space of ``small compact operators''). It is proved that ${\bf K}$ has the CSCP, using the second author's previous result that ${\bf K}_0$ has this property. A new proof is given for the result (due to E. Kirchberg) that ${\bf K}_0$ (and hence ${\bf K}$) fails the CSEP. It remains an open question if ${\bf K}$ has the MSEP; it is proved this is equivalent to whether ${\bf K}_0$ has this property. A new Banach space concept, Extendable Local Reflexivity (ELR), is introduced to study this problem. Further complements and open problems are discussed.

math.OA↗
arXiv1997

The ``maximal" tensor product of operator spaces

In analogy with the maximal tensor product of $C^*$-algebras, we define the ``maximal" tensor product $E_1\otimes_μE_2$ of two operator spaces $E_1$ and $E_2$ and we show that it can be identified completely isometrically with the sum of the two Haagerup tensor products: \ $E_1\otimes_h E_2 + E_2\otimes_h E_1$. Let $E$ be an $n$-dimensional operator space. As an application, we show that the equality $E^* \otimes_μE=E^* \otimes_{\rm min} E$ holds isometrically iff $E = R_n$ or $E=C_n$ (the row or column $n$-dimensional Hilbert spaces). Moreover, we show that if an operator space $E$ is such that, for any operator space $F$, we have $F\otimes_{\min} E=F\otimes_μ E$ isomorphically, then $E$ is completely isomorphic to either a row or a column Hilbert space.

math.FA↗