An introduction to nuclear space

This is a small book (48 pages) that contains a revised and extended version of the notes of seminar lectures given by Bessaga. The authors present a nice introduction to nuclear spaces (with all necessary preliminaries) based on Kolmogorov diameters. They consider only some of the most important topics of the theory of nuclear spaces, namely Kolmogorov diameters, nuclear operators, Mityagin's characterization of nuclear spaces, the theorem on absoluteness of bases in nuclear spaces, the uniqueness problem for bases (together with the theorem on quasiequivalence of regular bases), examples of nuclear Fréchet spaces without basis. Of course, many of the important topics in the theory of nuclear spaces are not even touched. Nevertheless, this book may be recommended to anyone who wants to study nuclear spaces, since it is practically independent of other sources and covers an essential part of the theory. Moreover, the authors do their best to help the reader: the proofs of all theorems are complete and the organization of the material is perfect

The absolutely continuous spectrum of one-dimensional Schr"odinger operators

This paper deals with general structural properties of one-dimensional Schr"odinger operators with some absolutely continuous spectrum. The basic result says that the omega limit points of the potential under the shift map are reflectionless on the support of the absolutely continuous part of the spectral measure. This implies an Oracle Theorem for such potentials and Denisov-Rakhmanov type theorems. In the discrete case, for Jacobi operators, these issues were discussed in my recent paper [19]. The treatment of the continuous case in the present paper depends on the same basic ideas.Comment: references added; a few very minor change

Compact convex sets in 2-dimensional asymmetric normed lattices

[EN] In this note, we study the geometric structure of compact convex sets in 2-dimensional asymmetric normed lattices. We prove that every q-compact convex set is strongly q-compact and we give a complete geometric description of the compact convex set with non empty interior in (R-2, q), where q is an asymmetric lattice norm.The first author has been supported by CONACYT (Mexico) under Grant 204028. The second author has been supported by the Ministerio de Economia y Competitividad (Spain) under Grant MTM2012-36740-C02-02.Jonard-Perez, N.; Sánchez Pérez, EA. (2016). Compact convex sets in 2-dimensional asymmetric normed lattices. Quaestiones Mathematicae. 39(1):73-82. https://doi.org/10.2989/16073606.2015.1023864S738239

Shift invariant preduals of ℓ₁(ℤ)

The Banach space ℓ<sub>1</sub>(ℤ) admits many non-isomorphic preduals, for example, C(K) for any compact countable space K, along with many more exotic Banach spaces. In this paper, we impose an extra condition: the predual must make the bilateral shift on ℓ<sub>1</sub>(ℤ) weak<sup>*</sup>-continuous. This is equivalent to making the natural convolution multiplication on ℓ<sub>1</sub>(ℤ) separately weak*-continuous and so turning ℓ<sub>1</sub>(ℤ) into a dual Banach algebra. We call such preduals <i>shift-invariant</i>. It is known that the only shift-invariant predual arising from the standard duality between C<sub>0</sub>(K) (for countable locally compact K) and ℓ<sub>1</sub>(ℤ) is c<sub>0</sub>(ℤ). We provide an explicit construction of an uncountable family of distinct preduals which do make the bilateral shift weak<sup>*</sup>-continuous. Using Szlenk index arguments, we show that merely as Banach spaces, these are all isomorphic to c<sub>0</sub>. We then build some theory to study such preduals, showing that they arise from certain semigroup compactifications of ℤ. This allows us to produce a large number of other examples, including non-isometric preduals, and preduals which are not Banach space isomorphic to c<sub>0</sub>

Normal systems over ANR's, rigid embeddings and nonseparable absorbing sets

Most of results of Bestvina and Mogilski [\textit{Characterizing certain incomplete infinite-dimensional absolute retracts}, Michigan Math. J. \textbf{33} (1986), 291--313] on strong $Z$-sets in ANR's and absorbing sets is generalized to nonseparable case. It is shown that if an ANR $X$ is locally homotopy dense embeddable in infinite-dimensional Hilbert manifolds and $w(U) = w(X)$ (where `$w$' is the topological weight) for each open nonempty subset $U$ of $X$,then $X$ itself is homotopy dense embeddable in a Hilbert manifold. It is also demonstrated that whenever $X$ is an AR, its weak product $W(X,*) = \{(x_n)_{n=1}^{\infty} \in X^{\omega}:\ x_n = * \textup{for almost all} n\}$ is homeomorphic to a pre-Hilbert space $E$ with $E \cong \Sigma E$. An intrinsic characterization of manifolds modelled on such pre-Hilbert spaces is given.Comment: 26 page

On linear operators and functors extending pseudometrics

For some pairs (X,A), where X is a metrizable topological space and A its closed subset, continuous, linear (i.e., additive and positive-homogeneous) operators extending metrics for A to metrics for X are constructed. They are defined by explicit analytic formulas, and also regarded as functors between certain categories. An essential role is played by "squeezed cones" related to the classical cone construction. The main result: if A is a nondegenerate absolute neighborhood retract for metric spaces, then continuous linear operators extending metrics always exist