arXiv subjects

K. Tyros

Publications and source records attributed to K. Tyros.

4 recordsLinked to original sources

arXiv2014

Spreading Models in Banach Space Theory

We extend the classical Brunel-Sucheston definition of the spreading model by introducing the $\mathcal{F}$-sequences $(x_s)_{s\in\mathcal{F}}$ in a Banach space and the plegma families in $\mathcal{F}$ where $\mathcal{F}$ is a regular thin family. The new concept yields a transfinite increasing hierarchy of classes of spreading sequences. We explore the corresponding theory and we present examples establishing this hierarchy and illustrating the limitation of the theory.

math.FA↗

arXiv2012

Higher Order Spreading Models

We introduce the higher order spreading models associated to a Banach space $X$. Their definition is based on $\ff$-sequences $(x_s)_{s\in\ff}$ with $\ff$ a regular thin family and the plegma families. We show that the higher order spreading models of a Banach space $X$ form an increasing transfinite hierarchy $(\mathcal{SM}_ξ(X))_{ξ<ω_1}$. Each $\mathcal{SM}_ξ(X)$ contains all spreading models generated by $\ff$-sequences $(x_s)_{s\in\ff}$ with order of $\ff$ equal to $ξ$. We also provide a study of the fundamental properties of the hierarchy.

math.FA↗

arXiv2011

Finite order spreading models

Extending the classical notion of the spreading model, the $k$-spreading models of a Banach space are introduced, for every $k\in\mathbb{N}$. The definition, which is based on the $k$-sequences and plegma families, reveals a new class of spreading sequences associated to a Banach space. Most of the results of the classical theory are stated and proved in the higher order setting. Moreover, new phenomena like the universality of the class of the 2-spreading models of $c_0$ and the composition property are established. As consequence, a problem concerning the structure of the $k$-iterated spreading models is solved.

math.FA↗

arXiv2010

A discretized approach to W.T. Gowers' game

We give an alternative proof of W. T. Gowers' theorem on block bases by reducing it to a discrete analogue on specific countable nets. We also give a Ramsey type result on k-tuples of block sequences in a normed linear space with a Schauder basis.

math.FA↗