Descriptive Set Theory and Applications
Doktorsavhandling, 2021
The systematic study of Polish spaces within the scope of Descriptive Set Theory furnishes the working mathematician with powerful techniques and illuminating insights. In this thesis, we start with a concise recapitulation of some classical aspects of Descriptive Set Theory which is followed by a succint review of topological groups, measures and some of their associated algebras.
The main application of these techniques contained in this thesis is the study of two families of closed subsets of a locally compact Polish group
G, namely U(G) - closed sets of uniqueness - and U0(G) - closed sets of extended uniqueness. We locate the descriptive set theoretic complexity
of these families, proving in particular that U(G) is \Pi_1^1-complete whenever G/\overline{[G,G]} is non-discrete, thereby extending the existing literature regarding the abelian case. En route, we establish some preservation results concerning sets of (extended) uniqueness and their operator theoretic counterparts. These results constitute a pivotal part in the arguments used and entail alternative proofs regarding the computation of the complexity of U(G) and U0(G) in some classes of the abelian case.
We study U(G) as a calibrated \Pi_1^1 \sigma-ideal of F(G) - for G amenable - and prove some criteria concerning necessary conditions for the inexistence of a Borel basis for U(G). These criteria allow us to retrieve information about G after examination of its subgroups or quotients. Furthermore, a sufficient condition for the inexistence of a Borel basis for U(G) is proven for the case when G is a product of compact (abelian or not) Polish groups
satisfying certain conditions.
Finally, we study objects associated with the point spectrum of linear bounded operators T\in L(X) acting on a separable Banach space X. We provide a characterization of reflexivity for Banach spaces with an unconditional basis : indeed such space X is reflexive if and only if for all closed subspaces Y\subset X;Z\subset X^{\ast} and T\in 2 L(Y); S\in 2 L(Z) it holds that the point spectra \sigma_p(T); \sigma_p(S) are Borel. We study the complexity of sets prescribed by eigenvalues and prove a stability criterion for Jamison sequences.
The main application of these techniques contained in this thesis is the study of two families of closed subsets of a locally compact Polish group
G, namely U(G) - closed sets of uniqueness - and U0(G) - closed sets of extended uniqueness. We locate the descriptive set theoretic complexity
of these families, proving in particular that U(G) is \Pi_1^1-complete whenever G/\overline{[G,G]} is non-discrete, thereby extending the existing literature regarding the abelian case. En route, we establish some preservation results concerning sets of (extended) uniqueness and their operator theoretic counterparts. These results constitute a pivotal part in the arguments used and entail alternative proofs regarding the computation of the complexity of U(G) and U0(G) in some classes of the abelian case.
We study U(G) as a calibrated \Pi_1^1 \sigma-ideal of F(G) - for G amenable - and prove some criteria concerning necessary conditions for the inexistence of a Borel basis for U(G). These criteria allow us to retrieve information about G after examination of its subgroups or quotients. Furthermore, a sufficient condition for the inexistence of a Borel basis for U(G) is proven for the case when G is a product of compact (abelian or not) Polish groups
satisfying certain conditions.
Finally, we study objects associated with the point spectrum of linear bounded operators T\in L(X) acting on a separable Banach space X. We provide a characterization of reflexivity for Banach spaces with an unconditional basis : indeed such space X is reflexive if and only if for all closed subspaces Y\subset X;Z\subset X^{\ast} and T\in 2 L(Y); S\in 2 L(Z) it holds that the point spectra \sigma_p(T); \sigma_p(S) are Borel. We study the complexity of sets prescribed by eigenvalues and prove a stability criterion for Jamison sequences.
Sets of Uniqueness
Operator U-sets
Thin Sets
Descriptive Set Theory,
Operator U0-sets
Point Spectrum
Reflexivity
Fourier Algebra
Jamison sequences
Harmonic Analysis
Pascal room
Opponent: Professor Jean Esterle, Institut de Mathématiques de Bordeaux, France
Författare
Joao Pedro Paulos
Chalmers, Matematiska vetenskaper, Analys och sannolikhetsteori
Descriptive set-theoretic aspects of closed sets of uniqueness in the non-abelian setting
Studia Mathematica,; Vol. 265(2022)p. 77-109
Artikel i vetenskaplig tidskrift
Stability of Jamison sequences under certain perturbations
North-Western European Journal of Mathematics,; Vol. 5(2019)p. 88-99
Artikel i vetenskaplig tidskrift
ON REFLEXIVITY AND POINT SPECTRUM
Real Analysis Exchange,; Vol. 47(2022)p. 167-178
Artikel i vetenskaplig tidskrift
What constitutes the truth of a mathematical claim ? Most likely, any first attempt to answer this question will appeal to the idea of a proof. A proof is, roughly speaking, a finite sequence of claims which we know to be true and that are related with each other within the system of logic we decided to adopt. In other words, a reasoning like : P1 is true since P2 is true and we can conclude P1 from P2; P2 is true because we can deduce it from P3 and P4 which we know to be true and so on. Consequently, in order to be able to derive anything at all, we need to start with some statements which are taken to be true a priori - the axioms. The systematic study of these issues is an important object of interest in Set Theory. Indeed, one can say that set theorists study the foundational aspects of the mathematical edifice. This is done towards the goal of discovering (or inventing) a robust Constitution which sustains the development of ideas with crucial applications in science. Moreover, a powerful language - enabling us to express certain intuitions and aesthetic urges - can thus be developed in a rigorous, reliable manner.
One can endow a set, i.e. a legitimate collection of objects like the set R of all real numbers, with an operation that satisfy certain properties - e.g. R equipped with addition. This is called a group. Groups are entities studied in Mathematics that crystallize the notion of symmetry. A topology is yet another structure that one can consider on a set. Crudely speaking, a topology prescribes the notion of nearness between elements of a set - e.g. R endowed with the usual distance. The study of these abstract objects has remarkably broad and profound applications outside the realm of pure Mathematics. One can combine in a compatible way these two structures - i.e. the group operation and the topology on a set. Topological groups is what emerges from this fruitful, rich interaction.
In this thesis, we study properties of sets associated with topological groups - namely the so called closed sets of (extended) uniqueness. The study of these sets has a long and illustrious history. In fact, the genesis of this research area lead to staggering ideas such as infinities of different sizes, igniting the revolution of Set Theory in the end of the 19th century. Around a century later, the application of set theoretic avoured techniques constituted a major source of new insights and solved open problems in the subject. The core of this thesis is thus focused in this successful interdisciplinary interaction. We extend some existing results regarding a notion of complexity of these sets.
Other topics are approached - still through a somewhat set theoretic lens. In particular, we compute the aforementioned notion of complexity of sets associated with linear bounded operators - i.e. infinite dimensional generalizations of simple geometric operations, like a translation - and we study how stable certain sequences - the so called Jamison sequences - are.
One can endow a set, i.e. a legitimate collection of objects like the set R of all real numbers, with an operation that satisfy certain properties - e.g. R equipped with addition. This is called a group. Groups are entities studied in Mathematics that crystallize the notion of symmetry. A topology is yet another structure that one can consider on a set. Crudely speaking, a topology prescribes the notion of nearness between elements of a set - e.g. R endowed with the usual distance. The study of these abstract objects has remarkably broad and profound applications outside the realm of pure Mathematics. One can combine in a compatible way these two structures - i.e. the group operation and the topology on a set. Topological groups is what emerges from this fruitful, rich interaction.
In this thesis, we study properties of sets associated with topological groups - namely the so called closed sets of (extended) uniqueness. The study of these sets has a long and illustrious history. In fact, the genesis of this research area lead to staggering ideas such as infinities of different sizes, igniting the revolution of Set Theory in the end of the 19th century. Around a century later, the application of set theoretic avoured techniques constituted a major source of new insights and solved open problems in the subject. The core of this thesis is thus focused in this successful interdisciplinary interaction. We extend some existing results regarding a notion of complexity of these sets.
Other topics are approached - still through a somewhat set theoretic lens. In particular, we compute the aforementioned notion of complexity of sets associated with linear bounded operators - i.e. infinite dimensional generalizations of simple geometric operations, like a translation - and we study how stable certain sequences - the so called Jamison sequences - are.
Ämneskategorier
Matematik
ISBN
978-91-7905-529-5
Doktorsavhandlingar vid Chalmers tekniska högskola. Ny serie: 4996
Utgivare
Chalmers