#
Descriptive Set Theory and Applications
Doctoral thesis, 2021

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

## Author

### Joao Pedro Paulos

Chalmers, Mathematical Sciences, Analysis and Probability Theory

### Descriptive set-theoretic aspects of closed sets of uniqueness in the non-abelian setting

Studia Mathematica,; Vol. 265(2022)p. 77-109

**Journal article**

### Stability of Jamison sequences under certain perturbations

North-Western European Journal of Mathematics,; Vol. 5(2019)p. 88-99

**Journal article**

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.

### Subject Categories

Mathematics

### ISBN

978-91-7905-529-5

Doktorsavhandlingar vid Chalmers tekniska högskola. Ny serie: 4996

### Publisher

Chalmers