Jarosław Swaczyna
Jarosław Swaczyna: Zoo of ideal Schauder bases
24/11/23 12:02
Tuesday, November 28, 2023 17:00
Location: room 601, Mathematical Institute, University of Wroclaw
Speaker: Jarosław Swaczyna (Łódź University of Technology)
Title: Zoo of ideal Schauder bases
Abstract: Given a Banach space \(X\), sequence \((e_n)\) of its elements and an ideal \(I\) on natural numbers we say that \((e_n)\) is an \(I\)-Schauder base if for every \(x \in X\) there exists unique sequence of scalars \(\alpha_n\) such that series of \(\alpha_n e_n\) is \(I\)-convergent to \(X\). In such a case one may also consider coordinate functionals \(e_n^\star\). About ten years ago Kadets asked if those functionals are necessarily continuous at least for some nice ideals, e.g. the ideal of sets of density zero. During my talk I will present an answer to this question obtained jointly with Tomasz Kania and Noe de Rancourt. I will also present some examples of ideal Schauder bases which are not the classical ones. Second part will be based on ongoing work with Adam Kwela.
Location: room 601, Mathematical Institute, University of Wroclaw
Speaker: Jarosław Swaczyna (Łódź University of Technology)
Title: Zoo of ideal Schauder bases
Abstract: Given a Banach space \(X\), sequence \((e_n)\) of its elements and an ideal \(I\) on natural numbers we say that \((e_n)\) is an \(I\)-Schauder base if for every \(x \in X\) there exists unique sequence of scalars \(\alpha_n\) such that series of \(\alpha_n e_n\) is \(I\)-convergent to \(X\). In such a case one may also consider coordinate functionals \(e_n^\star\). About ten years ago Kadets asked if those functionals are necessarily continuous at least for some nice ideals, e.g. the ideal of sets of density zero. During my talk I will present an answer to this question obtained jointly with Tomasz Kania and Noe de Rancourt. I will also present some examples of ideal Schauder bases which are not the classical ones. Second part will be based on ongoing work with Adam Kwela.
Jarosław Swaczyna: Haar-small sets
18/05/17 10:55
Tuesday, May 23, 2017 17:15
Room: D1-215
Speaker: Jarosław Swaczyna
Title: Haar-small sets
Abstract. In locally compact Polish groups there is a very natural \(\sigma\)-ideal of null sets with respect to Haar-measure. In non locally compact groups there is no Haar measure, however Christensen introduced a notion of Haar-null sets which is an analogue of locally compact case. In 2013 Darji introduced a similar notion of Haar-meager sets. During my talk I will present some equivalent definition of Haar-null sets which leads us to joint generalization of those notions. This is joint work with T. Banakh, Sz. Głąb and E. Jabłońska.
Room: D1-215
Speaker: Jarosław Swaczyna
Title: Haar-small sets
Abstract. In locally compact Polish groups there is a very natural \(\sigma\)-ideal of null sets with respect to Haar-measure. In non locally compact groups there is no Haar measure, however Christensen introduced a notion of Haar-null sets which is an analogue of locally compact case. In 2013 Darji introduced a similar notion of Haar-meager sets. During my talk I will present some equivalent definition of Haar-null sets which leads us to joint generalization of those notions. This is joint work with T. Banakh, Sz. Głąb and E. Jabłońska.
Jarosław Swaczyna: Generalized densities of subsets of natural numbers and associated ideals
15/05/15 19:36
Tuesday, May 19, 2015 17:15
Room: D1-215
Speaker: Jarosław Swaczyna
Title: Generalized densities of subsets of natural numbers and associated ideals
Abstract. Let \(g: \omega \rightarrow [0, \infty)\). We say that \(A \subset \omega\) has \(g\)-density zero, if \(\lim_{n \rightarrow \infty} \frac{A \cap n}{g(n)} = 0\). It is an easy observation that family of \(g\)-density zero sets is an ideal.
I will discuss some properties of ideals obtained this way (among others, I will show that they can be generated using Solecki's submeasures). I will then examine inclusions between ideals obtained for different functions \(g\).
I will also discuss connections between our ideals, "density-like" ideals and Erdos-Ulam ideals. I will present joint results with M. Balcerzak, P. Das and M. Filipczak.
Room: D1-215
Speaker: Jarosław Swaczyna
Title: Generalized densities of subsets of natural numbers and associated ideals
Abstract. Let \(g: \omega \rightarrow [0, \infty)\). We say that \(A \subset \omega\) has \(g\)-density zero, if \(\lim_{n \rightarrow \infty} \frac{A \cap n}{g(n)} = 0\). It is an easy observation that family of \(g\)-density zero sets is an ideal.
I will discuss some properties of ideals obtained this way (among others, I will show that they can be generated using Solecki's submeasures). I will then examine inclusions between ideals obtained for different functions \(g\).
I will also discuss connections between our ideals, "density-like" ideals and Erdos-Ulam ideals. I will present joint results with M. Balcerzak, P. Das and M. Filipczak.