Linear polarization constants of Hilbert spaces

This paper has been motivated by previous work on estimating lower bounds for the norms of homogeneous polynomials which are products of linear forms. The purpose of this work is to investigate the so-called nth (linear) polarization constant cn(X) of a finite-dimensional Banach space X, and in particular of a Hilbert space. Note that cn(X) is an isometric invariant of the space. It has been proved by J. Arias-de-Reyna [Linear Algebra Appl. 285 (1998) 395-408] that if H is a complex Hilbert space of dimension at least n, then cn(H)=nn/2. The same value of cn(H) for real Hilbert spaces is only conjectured, but estimates were obtained in many cases. In particular, it is known that the nth (linear) polarization constant of a d-dimensional real or complex Hilbert space H is of the approximate order dn/2, for n large enough, and also an integral form of the asymptotic quantity c(H), that is the (linear) polarization constant of the Hilbert space H, where dimH=d, was obtained together with an explicit form for real spaces. Here we present another proof, we find the explicit form even for complex spaces, and we elaborate further on the values of cn(H) and c(H). In particular, we answer a question raised by J.C. García-Vázquez and R. Villa [Mathematika 46 (1999) 315-322]. Also, we prove the conjectured cn(H)=nn/2 for real Hilbert spaces of dimension n?5. A few further estimates have been also derived.     

Linear homeomorphisms of spaces of continuous functions on long Sorgenfrey lines

We carry out a linear homeomorphic classification of the spaces of continuous functions on the long Sorgenfrey lines S_α, where a is an arbitrary ordinal. The spaces of continuous functions are endowed with the topology of pointwise convergence and denoted by C_p(S_α).     

Lipschitz homeomorphisms of the Hilbert cube

We study the question whether the Hilbert cube Q is Lipschitz homogeneous. The answer depends on the metric of Q. For example, setting d(x,y)=sup_j|x_j-y_j|/j we obtain a Lipschitz homogeneous metric, but if the last j is replaced by j!, the answer is negative.     

广义近似空间的粗糙同胚与拓扑同胚

对广义近似空间之间的映射引入并刻画了粗糙连续性和拓扑连续性,探讨了他们的性质及相互关系.证明了两个粗糙连续映射的复合还是粗糙连续的,每个粗糙连续的映射都是拓扑连续的.在此基础上引入了粗糙同胚性质和拓扑同胚性质的概念,证明了拓扑同胚性质均为粗糙同胚性质并考察了广义近似空间的诸如分离性、连通性、紧性等的粗糙同胚不变性和拓扑同胚不变性.     

Rigged Hilbert spaces and contractive families of Hilbert spaces

The existence of a rigged Hilbert space whose extreme spaces are, respectively, the projective and the inductive limit of a directed contractive family of Hilbert spaces is investigated. It is proved that, when it exists, this rigged Hilbert space is the same as the canonical rigged Hilbert space associated to a family of closable operators in the central Hilbert space.     

Positively regular homeomorphisms of Euclidean spaces

A homeomorphism of Rⁿ onto itself is called positively regular (or EC⁺) iff its family of non-negative iterates is pointwise equicontinuous. For EC⁺ homeomorphism of Rⁿ such that some point of Rⁿ has bounded positive semi-orbit, the nucleus M is defined, and the following theorems are proved.Theorem 1. If such a homeomorphism h:Rⁿ→Rⁿ has compact nucleus M, then M is a fully invariant compact AR. Further, for n≠4,5,h:Rⁿ/M→Rⁿ/M is conjugate to a contraction on Rⁿ.Theorem 2. In Rⁿ,n≠4,5,M compact iff there existsa disk D such that h(D)?IntD.Theorem 3. In R², either M is a disk and h|M is a rotation, or h|M is periodic. The relationship between M and the irregular set of ? is also studied.     

Positively expansive homeomorphisms on compact metric spaces

We give a new and simple proof to show that no homeomorphism of infinite compact metric spaces is positively expansive.     

Sub-Bergman Hilbert spaces

In this paper we analyze sub-Bergman Hilbert spaces in the unit disk associated with finite Blaschke products. We also analyze their analogues in weighted Bergman spaces.     

Ultrametric spaces and related Hilbert spaces

We present simple proofs of the possibility of embedding ultrametric spaces in Hilbert spaces. The main part of the paper deals with ultrametric spaces that we call totally infinite spaces. Related Hilbert spaces, automorphisms of totally infinite spaces, and the corresponding linear operators are considered. Transplated fromMatematicheskie Zametki, Vol. 62, No. 2, pp. 223–237, August, 1997. Translated by V. E. Nazaikinskii     

Subgroups of Hilbert spaces

The author was supported by the Alexander von Humboldt-Stiftung     

Imaginaries in Hilbert spaces

We characterise imaginaries (up to interdefinability) in Hilbert spaces using a Galois theory for compact unitary groups.The authors would like to thank Frank Wagner and the CIRM for their hospitality during the Simpleton 2002 meeting during which the discussions that led to this paper took place.The first author would like to thank Ilan Hirshberg for a few important hintsAt the time of the writing of this paper, the first author was a graduate student with the Équipe de logique mathématique of Université Paris VIIMathematics Subject Classification (2000): 03C45, 03C95     

Multishifts in Hilbert spaces

We introduce and study a multishift structure in a Hilbert space. This structure is a noncommutative analog of the (simple one-sided) shift operator, well known in function theory and functional analysis. Subspaces invariant under the multishift are described. A theorem on the factorization into an inner and an outer factor is established for operators commuting with the multishift.__________Translated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 39, No. 1, pp. 69–81, 2005Original Russian Text Copyright © by P. A. TerekhinSupported by the RFBR under grant No. 03-01-00390, the program Leading Scientific Schools of the Russian Federation under grant No. NSh-1295.2003.1, and the INTAS under grant No. 99-00089.Translated by V. E. Nazaikinskii     

Global homeomorphisms and covering projections on metric spaces

For a large class of metric spaces with nice local structure, which includes Banach–Finsler manifolds and geodesic spaces of curvature bounded above, we give sufficient conditions for a local homeomorphism to be a covering projection. We first obtain a general condition in terms of a path continuation property. As a consequence, we deduce several conditions in terms of path- liftings involving a generalized derivative, and in particular we obtain an extension of Hadamard global inversion theorem in this context. Next we prove that, in the case of quasi-isometric mappings, some of these sufficient conditions are also necessary. Finally, we give an application to the existence of global implicit functions. O. Gutú and J. A. Jaramillo were supported in part by D.G.E.S. (Spain) Grant BFM2003-06420.