Jarosław Swaczyna

Jarosław Swaczyna: Zoo of ideal Schauder bases

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 (en) of its elements and an ideal I on natural numbers we say that (en) is an I-Schauder base if for every xX there exists unique sequence of scalars αn such that series of αnen is I-convergent to X. In such a case one may also consider coordinate functionals en. 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

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 σ-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

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:ω[0,). We say that Aω has g-density zero, if limnAng(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.