Advanced Courses of Mathematical Analysis III

PREFACE

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CONTENTS

The following sections are included:

• Introduction

• Hyperbolic dynamics

• Superattracting dynamics

• Parabolic dynamics

• Elliptic dynamics

• References

The following sections are included:

• Notation and preliminaries

• Methods and applications

• References

The following sections are included:

• Discrete translates, generating functions

• Connections with density of exponentials

• The problem in L¹(ℝ) and its connections with the spectral radious problem

• Relation with quasianalytic classes

• The spectral sets for L¹(ℝ)

• The generators for L¹(ℝ)

• The restricted problem

• References

Many classical results about compactness in functional analysis can be derived from suitable inequalities involving distances to spaces of continuous or Baire one functions: this approach gives an extra insight to the classical results as well as triggers a number of open questions in different exciting research branches. We exhibit here, for instance, quantitative versions of Grothendieck's characterization of weak compactness in spaces C(K) and also of the Eberlein–Šmulyan and Krein–Šmulyan theorems. The above results specialized in Banach spaces lead to several equivalent measures of non-weak compactness. In a different direction we envisage a method to measure the distance from a function f ∈ ℝ^X to B₁(X) — space of Baire one functions on X — which allows us to obtain, when X is Polish, a quantitative version of the well-known Rosenthal's result stating that in B₁(X) the pointwise relatively countably compact sets are pointwise compact. Other results and applications are commented too.

We will pose several situations in analysis where some classes of smooth functions play a fundamental role. In connection with the study of Laplace equation, we shall analyze the behavior of the fractional integral operator on L^p spaces, where BMO and Lipschitz spaces arise in a natural way. As a generalization, we will present and study a family of spaces introduced by Spanne. In particular we will be interested in identifying those members of the family containing only continuous functions. Finally we shall present a brief description of Besov spaces and their connection with a problem of non-linear approximation of a function by its wavelet expansion.

The aim of this talk is to study the domination problem for the class of strictly singular operators and other related operator classes. We also discuss the strict singularity of inclusions between rearrangement invariant function spaces.

The following sections are included:

• Introduction

• The origin of the Hahn – Banach Theorem

• Helly

• The "Landmark": Helly's 1921 article

• Helly's technique

• The complex case

• Intersection properties

• References

Our goal is to explore the structure of the "small" subspaces of L_p mainly for 2 < p < ∞, discussing older classical results and ultimately presenting some new results of [12]. We will review first some Banach space basics. By L_p we shall mean L_p[0,1], under Lebesgue measure m…

In this talk we review properties of hypercyclic operators, revisit some well known results, and present several problems, some of them in the new classes of dual hypercyclic operators and frequently hypercyclic operators.

The following sections are included:

• Operator spaces and completely bounded maps

• The fundamental theorem and some consequences

• Tensor products of operator spaces

• Applications to operator algebra theory

• Applications to harmonic analysis

• Applications to Probability Theory

• References

The following sections are included:

• Introduction

• Social equilibrium

• From social equilibrium to competitive equilibrium

• Equilibrium and effiency

• Final comments

• References

We give here a survey on several open problems and several related, mostly recent, results concerning ideals in F-algebras and its subclasses (B₀-algebras and m-convex B₀-algebras).