arXiv subjects
Publications and source records attributed to M. I. Ostrovskii.
The main purpose of the paper is to develop an approach to evaluation or estimation of the spanning tree congestion of planar graphs. This approach is used to evaluate the spanning tree congestion of triangular grids.
A new technique for proving fixed point theorems for families of holomorphic transformations of operator balls is developed. One of these theorems is used to show that a bounded representation in a real or complex Hilbert space is orthogonalizable or unitarizable (that is similar to an orthogonal or unitary representation), respectively, provided the representation has an invariant indefinite quadratic form with finitely many negative squares.
The main purpose of the paper is to find some expansion properties of locally finite metric spaces which do not embed coarsely into a Hilbert space. The obtained result is used to show that infinite locally finite graphs excluding a minor embed coarsely into a Hilbert space. In an appendix a direct proof of the latter result is given.
Let $K$ be an absolutely convex infinite-dimensional compact in a Banach space $\mathcal{X}$. The set of all bounded linear operators $T$ on $\mathcal{X}$ satisfying $TK\supset K$ is denoted by $G(K)$. Our starting point is the study of the closure $WG(K)$ of $G(K)$ in the weak operator topology. We prove that $WG(K)$ contains the algebra of all operators leaving $\overline{\lin(K)}$ invariant. More precise results are obtained in terms of the Kolmogorov $n$-widths of the compact $K$. The obtained results are used in the study of operator ranges and operator equations.
A necessary and sufficient condition for existence of a Banach space with a finite dimensional decomposition but without the $π$-property in terms of norms of compositions of projections is found.
Let $B_Y$ denote the unit ball of a normed linear space $Y$. A symmetric, bounded, closed, convex set $A$ in a finite dimensional normed linear space $X$ is called a {\it sufficient enlargement} for $X$ if, for an arbitrary isometric embedding of $X$ into a Banach space $Y$, there exists a linear projection $P:Y\to X$ such that $P(B_Y)\subset A$. The main results of the paper: {\bf (1)} Each minimal-volume sufficient enlargement is linearly equivalent to a zonotope spanned by multiples of columns of a totally unimodular matrix. {\bf (2)} If a finite dimensional normed linear space has a minimal-volume sufficient enlargement which is not a parallelepiped, then it contains a two-dimensional subspace whose unit ball is linearly equivalent to a regular hexagon.
We show that the open unit ball of the space of operators from a finite dimensional Hilbert space into a separable Hilbert space (we call it "operator ball") has a restricted form of normal structure if we endow it with a hyperbolic metric (which is an analogue of the standard hyperbolic metric on the unit disc in the complex plane). We use this result to get a fixed point theorem for groups of biholomorphic automorphisms of the operator ball. The fixed point theorem is used to show that a bounded representation in a separable Hilbert space which has an invariant indefinite quadratic form with finitely many negative squares is unitarizable (equivalent to a unitary representation). We apply this result to find dual pairs of invariant subspaces in Pontryagin spaces. In the appendix we present results of Itai Shafrir about hyperbolic metrics on the operator ball.
The main purposes of this paper are (1) To survey the area of coarse embeddability of metric spaces into Banach spaces, and, in particular, coarse embeddability of different Banach spaces into each other; (2) To present new results on the problems: (a) Whether coarse non-embeddability into $\ell_2$ implies presence of expander-like structures? (b) To what extent $\ell_2$ is the most difficult space to embed into?
Let X be a Banach space. Given a subset A of the dual space X* denote by $A_{(1)}$ the weak* sequential closure of A, i.e., the set of all limits of weak*-convergent sequences in A. The study of weak* sequential closures of linear subspaces of the duals of separable Banach spaces was initiated by S.Banach. The first results of this study were presented in the appendix to his book "Theorie des operations lineaires" (1932). It is natural to suppose that the reason for studying weak* sequential closures by S. Banach and S. Mazurkiewicz was the lack of acquaintance of S. Banach and his school with concepts of general topology. Although the name "General topology" was introduced later, the subject did already existed. F. Hausdorff introduced topological spaces in his book published in 1914, Alexandroff-Urysohn (1924) studied compactness, and A.Tychonoff published his theorem on compactness of products in 1929. Also J.von Neumann introduced the notion of a weak topology in his paper published in 1929. Using the notions of a topological space and the Tychonoff theorem, more elegant treatment of weak and weak* topologies, and the duality of Banach spaces was developed by L.Alaoglu, N.Bourbaki and S.Kakutani (1938-1940). Nevertheless, an "old-fashioned" treatment of S.Banach still attracts attention. It happens because the "sequential" approach is very useful in several contexts. The main purpose of the paper is to describe the history of this direction of research and to give an up-to-date (1999) survey of the results on weak* sequential closures and their applications.
Definition. A symmetric with respect to 0 bounded closed convex set A in a finite dimensional normed space X is called a sufficient enlargement for X (or of B(X)) if for arbitrary isometric embedding of X into a Banach space Y there exists a projection P:Y\to X such that P(B(Y)) is a subset of A (by B(X) we denote the unit ball). The notion of sufficient enlargement is implicit in the paper: B.Grunbaum, Projection constants, Trans. Amer. Math. Soc. 95 (1960) 451--465. It was explicilty introduced by the author in: M.I.Ostrovskii, Generalization of projection constants: sufficient enlargements, Extracta Math., 11 (1996), 466--474. The main purpose of the present paper is to continue investigation of sufficient enlargements started in the papers cited above. In particular the author investigate sufficient enlargements whose support functions are in some directions close to those of the unit ball of the space, sufficient enlargements of minimal volume, sufficient enlargements for euclidean spaces.
We consider real spaces only. Definition. An operator $T:X\to Y$ between Banach spaces $X$ and $Y$ is called a Hahn-Banach operator if for every isometric embedding of the space $X$ into a Banach space $Z$ there exists a norm-preserving extension $\tilde T$ of $T$ to $Z$. A geometric property of Hahn-Banach operators of finite rank acting between finite-dimensional normed spaces is found. This property is used to characterize pairs of finite-dimensional normed spaces $(X,Y)$ such that there exists a Hahn-Banach operator $T:X\to Y$ of rank $k$. The latter result is a generalization of a recent result due to B.L. Chalmers and B. Shekhtman.