Norm-attaining functionals onC(Q, X)

Functionals (vector measures) defined on the spaceC(Q, X) of continuous abstract functions (whereQ is a compact Hausdorff space andX is a Banach space) and attaining their norm on the unit sphere are considered. A characterization of such functionals is given in terms of the Radon-Nikodym derivative of the vector measure with respect to the variation of the measure and in terms of analogs of the derivative. Applications to the characterization of finite-codimensional subspaces with the best approximation property are given. Similar results are obtained for the spaceB(Q, Σ, X) of uniform limits of simple functions. Translated fromMatematicheskie Zametki, Vol. 61, No. 1, pp. 45–56, January, 1997. Translated by V. E. Nazaikinskii     

Alexandroff's Double Arrow Compact Space and Approximation Theory

A compact space Q similar to the compact space known as Alexandroff's double arrow space is constructed. It is shown that the real space C(Q) has no Chebyshev subspaces of codimension >1, but the complex space C(Q) has such subspaces.     

On existence subspaces inC(Q)

The property of a space to be an existence subspace is studied for subspacesE ofC(Q) such that eitherE orC(Q)/E is a Lindenstrauss space. For a Chebyshev subspaceL⊂C(Q)₁ an analytic representation of the nearest element in terms of the annihilatorL ^┴ is obtained. Translated fromMatematicheskie Zametki, Vol. 65, No. 5, pp. 726–737, May, 1999.     

Metric projection onto finite-dimensional subspaces of C and L

In the space C(Q) of real functions that are continuous on the compact set Q, a finite-dimensional subspace P will have a uniformly continuous metric projection if and only if Q is a finite sum of compact sets Qi, and either P is on each Qi a one-dimensional Chebyshev space, or x(t)≡0 for any x belonging to P. The metric projection onto any finite-dimensional subspace of the space L[a, b] of real integrable functions is not uniformly continuous.     

Existence of best approximation elements inC(Q,X)

Generalizing the result of A. L. Garkavi (the caseX = ?) and his own previous result concerningX = ?), the author characterizes the existence subspaces of finite codimension in the spaceC(Q, X) of continuous functions on a bicompact spaceQ with values in a Banach spaceX, under some assumptions concerningX. Under the same assumptions, it is proved that in the space of uniform limits of simple functions, each subspace of the form $$\left\{ {g \in B:\smallint _Q \left\langle {g(t),d\mu _i } \right\rangle = 0,i = 1,...,n} \right\},$$ whereμ _i ∈ C(Q, X)^* are vector measures of regular bounded variation, is an existence subspace (the integral is understood in the sense of Gavurin).     

Topology of the hyperspace of convex bodies of constant width

The hyperspace of all convex bodies of constant width in Euclidean spaceR ⁿ ,n≥2, is proved to be homeomorphic to a contractibleQ-manifold (Q denotes the Hilbert cube). The proof makes use of an explicitly constructed retraction of the entire hyperspace of convex bodies on the hyperspace of convex bodies of constant width. Translated fromMaternaticheskie Zametki, Vol. 62, No. 6, pp. 813–819, December, 1997 Translated by V. N. Dubrovsky     

On based-free actions of compact lie groups on the hilbert cube

It is proved that a based-free action α of a given compact Lie groupG on the Hilbert cubeQ is equivalent to the standard based-free action σ if and only if the orbit spaceQ ₀/α of the free partQ ₀=Q* is aQ-manifold having the proper homotopy type of the orbit spaceQ ₀/σ. The existence of an equivariant retraction (Q ₀, σ)→(Q ₀, α) is established. It is proved that for any TikhonovG-spaceX the family of all equivariant mapsX→ conG separates the points and the closed sets inX. Translated fromMatematicheskie Zametki, Vol. 65, No. 2, pp. 163–174, February, 1999.     

On Copositive Approximation in Spaces of Continuous Functions I:The Alternation Property of Copositive Approximation

In this paper the author writes a simple characterization for the best copositive approximation to elements of C(Q) by elements of finite dimensional strict Chebyshev subspaces of C(Q) in the case when Q is any compact subset of real numbers. At the end of the paper the author applies this result for different classes of Q.     

On the countable definability of sets

LetQ be a connected set in ^p . Denote byD[Q] the set of all domains containingQ, and letW(Q) be the set of all convex domains fromD[Q]. We present tests for classesD[Q] andW(Q) (in the case whenQ is convex for the last one) to have a countable basis. The results are expressed in terms of properties of the boundary FrQ of the setQ.Translated fromMatematicheskie Zametki, Vol. 59, No. 3, pp. 382–395, March, 1996.This research was partially supported by the Russian Foundation for Basic Research under grant No. 93-011-242.     

On spaces of nearexistence

The notion of subspace of nearexistence is introduced. In particular, it is proved that ifQ is a countable compact set, then any subspaceL C(Q), dimL=codimL=+, can be approximated by subspaces of nearexistence.Translated fromMatematicheskie Zametki, Vol. 60, No. 2, pp. 278–287, August, 1996.This work was supported by the Fundamental Problems of Mathematics and Mechanics Foundation under grant No. 1.1.64.     

A Banach space in which all compact sets, but not all bounded sets, admit Chebyshev centers

Given a Banach spaceX, letc ₀(X) be the space of all null sequences inX (equipped with the supremum norm). We show that: 1) each compact set inc ₀(X) admits a (Chebyshev) center iff each compact set inX admits a center; 2) forX satisfying a certain condition (Q), each bounded set inc ₀(X) admits a center iffX is quasi uniformly rotund. We construct a Banach spaceX such that the compact subsets ofX admit centers,X satisfies the condition (Q) andX is not quasi uniformly rotund. It follows that the Banach spaceE=c ₀(X) has the property from the title. Eine überarbeitete Fassung ging am 4. 7. 2001 ein     

Products of orthogonal projections as Carleman operators

In this paper, we consider the product of two orthogonal projectionsP andQ on a separable, infinite dimensional Hilbert spaceH. For the operatorQP, there holds the dichotomy:QP is either a Carleman operator or a semi-Fredholm operator with finite defect. Both cases are characterized in terms of the dimensions of the ranges and null spaces ofP andQ and some of their intersections. This extends the case, whereP andQ are the special projections onto the subspaces of time- and band-limited functions inL ²() resp., first considered by Slepian, Pollak and Landau.     

Two-dimensional complex tori with multiplication by $\sqrt {d}$

We give an elementary argument for the well known fact that the endomorphism algebra End(A)?\Bbb Q {\rm {End}}(A)\otimes {\Bbb Q } of a simple complex abelian surface A can neither be an imaginary quadratic field nor a definite quaternion algebra. Another consequence of our argument is that a two-dimensional complex torus T with \Bbb Q (?d)\hookrightarrow End_{\Bbb Q} (T){\Bbb Q }(\sqrt {d})\hookrightarrow {\rm{End_{{\Bbb Q }}}}(T) where \Bbb Q (?d){\Bbb Q }(\sqrt {d}) is real quadratic, is algebraic.     

Complexity of the decidability of the unquantified set theory with a rank operator

The unquantified set theory MLSR containing the symbols ∪, ∖, ≠, ∈,R (R(x) is interpreted as a rank ofx) is considered. It is proved that there exists an algorithm which for any formulaQ of the MLSR theory decides whetherQ is true or not using the spacec|Q|³ (|Q| is the length ofQ).     

Picturing the lattice of invariant subspaces of a nilpotent complex matrix

Let Q be a nilpotent transformation acting on a finite-dimensional complex vector space. A method is given by which a diagrammatic representation of Lat Q, the lattice of invariant subspaces of Q, can be obtained. Basically the method consists of adding to the Hasse diagram of Hyperlat Q the finite lattice of hyperinvariant subspaces of Q in a specified way. Some applications are given.     

Lattice of subalgebras of the ring of continuous functions and Hewitt spaces

The latticeA(X) of all possible subalgebras of the ring of all continuous ℝ-valued functions defined on an ℝ-separated spaceX is considered. A topological space is said to be a Hewitt space if it is homeomorphic to a closed subspace of a Tychonoff power of the real line ℝ. The main achievement of the paper is the proof of the fact that any Hewitt spaceX is determined by the latticeA(X). An original technique of minimal and maximal subalgebras is applied. It is shown that the latticeA(X) is regular if and only ifX contains at most two points. Translated fromMatematicheskie Zametki, Vol. 62, No. 5, pp. 687–693, November, 1997. Translated by A. I. Shtern     

Чебыщевские подпространства в пространстве ХардиH 1

A subspaceY of a Banach spaceX is called a Chebyshev one if for everyx∈X there exists a unique elementP _Y (x) inY of best approximation. In this paper, necessary and sufficient conditions are obtained in order that certain classes of subspacesY of the Hardy spaceH ¹=H ¹ (|z|<1) be Chebyshev ones, and also the properties of the operatorP _Y are studied. These results show that the theory of Chebyshev subspaces inH ¹ differs sharply from the corresponding theory inL ¹(C) of complex-valued functions defined and integrable on the unit circleC:|z|=1. For example, it is proved that inH ¹ there exist sufficiently many Chebyshev subspaces of finite dimension or co-dimension (while inL ¹(C) there are no Chebyshev subspaces of finite dimension or co-dimension). Besides, it turned out that the collection of the Chebyshev subspacesY with a linear operatorP _Y inH ¹ (in contrast toL ¹(C)) is exhausted by that minimum which is necessary for any Banach space.     

On minimal,strongly proximal actions of locally compact groups

Minimal, strongly proximal actions of locally compact groups on compact spaces, also known asboundary actions, were introduced by Furstenberg in the study of Lie groups. In particular, the action of a semi-simple real Lie groupG on homogeneous spacesG/Q, whereQ ⊂G is a parabolic subgroup, are boundary actions. Countable discrete groups admit a wide variety of boundary actions. In this note we show that ifX is a compact manifold with a faithful boundary action of some locally compact groupH, then (under some mild regularity assumption) the groupH, the spaceX, and the action split into a direct product of a semi-simple Lie groupG acting onG/Q and a boundary action of a discrete countable group. The author was partially supported by NSF grants DMS-0049069, 0094245 and GIF grant G-454-213.06/95.