Author: Tomasz Kania

Continuity of coordinate functionals for filter Schauder basis without Large Cardinals

During my talk in November, I presented joint results with Tomasz Kania about using the large cardinals to positively answer the Kadets problem about coontinuity of coordinate functionals related to filter Schauder basis. This time I will present results obtained jointly with Tomasz Kania and Noe de Rancourt, which gives a positive answer to Kadets’ question in ZFC for analytic filters. I will also show that each filter basis with continuous coordinate functionals is also a basis with respect to some analytic filter.

Link to the talk.

Leave a Comment

Surjective Isometries of Banach sequence spaces: a survey

Beginning with the work of Banach on \(L_p\) spaces there are numerous papers characterizing the surjective isometries on various classical and non-classical Banach spaces. In this talk, we will give a broad overview of this work including the spaces of James, Schreier, and Tsirelson, and state several interesting open problems.

Leave a Comment

Guarded Fraïssé Banach spaces

Fraïssé theory, originally developed in the context of model theory, establishes a bijective correspondence between classes of finite structures having good amalgamation properties and so-called Fraïssé structures, i.e. countable structures satisfying a strong homogeneity property. This correspondence has recently been extended to Banach spaces by Ferenczi, Lopez-Abad, Mbombo et Todorcevic, who proved that the spaces \(L_p\), \(1 \leq p \neq 4, 6, 8, … < \infty\), and the Gurarij space, are Fraïssé. They asked whether those are the only examples; this question is related to Mazur’s rotation problem.

In a work in progress with Marek Cúth and Michal Doucha, we introduce a weak version of the Fraïssé correspondence, the so-called guarded Fraïssé correspondence, extending results obtained by Krawczyk and Kubiś in the discrete setting. We prove that a separable Banach space is guarded Fraïssé if and only if its isometry class is \(G_\delta\) for a natural topological coding of separable Banach spaces introduced by Cúth, Doležal, Doucha et Kurka. This links to the above-mentioned question of Ferenczi, Lopez-Abad, Mbombo et Todorcevic with descriptive set theory of Banach
spaces.

I will present those results and their links with some questions in continuous logic. I will also discuss the existence of new examples of guarded Fraïssé Banach spaces.

Leave a Comment

On a forgotten theorem of Pełczyński

Link to the talk

Leave a Comment

Some remarks about ‘‘Lipschitz-free operators’’

If M is a metric space, then the so-called Lipschitz-free space over M, usually denoted F(M), is a Banach space which is built around M in such a way that – M is isometric to a subset of F(M); – Lipschitz maps from M into any other Banach space X uniquely extend to bounded linear operators from F(M) into X. An interesting feature of Lipschitz-free spaces is that every Lipschitz map between two metric spaces M and N can be ‘‘linearised’’  in such a way that it becomes a bounded linear operator between the free spaces F(M) and F(N). We refer to these linearisations as ‘‘Lipschitz-free operators’’, or simply ‘‘Lipschitz operators’’. In this talk, we will study how the properties of the Lipschitz maps and their linearisations are related. After a few simple observations, we will mainly focus on some dynamical properties, compactness properties, and injectivity. This talk is based on ongoing works joint with Arafat Abbar (Marne-la-Vallée) and Clément Coine (Caen); Luis García-Lirola (Zaragoza) and Antonín Procházka (Besançon).

Leave a Comment

Interpolation of isomorphisms and Fredholm operators

We will discuss the interpolation of isomorphisms and Fredholm operators acting between Banach spaces generated by abstract interpolation methods. We will present general results regarding the stability of interpolated isomorphisms and the uniqueness of inverse operators between scales of interpolation spaces. We will show applications to PDE’s. The talk is based on the joint works with Irina Asekritova and Natan Kruglyak.

Link to the talk.

Leave a Comment

A complemented subspace of a C(K)-space which is not a C(K)-space

We present a construction of two separable compacta K and L such that C(L) is a direct sum of C(K) and some Banach space X which is not isomorphic to a space of continuous functions. Joint work with Alberto Salguero Alarcon (Badajoz).

Link to the talk.

Leave a Comment

Admissibility of Fréchet spaces

Let \(E\) be a Hausdorff topological vector space. A subset \( Z \) of \( E \) is said to be admissible if for every compact subset \( K \) of \( Z \) and for every neighbourhood \( V \) of zero in \( E \) there exists a continuous mapping \( H\colon K → Z \) such that \( \dim({\rm span} [H(K)])< \infty \) and \( x − Hx \in V \) for every \( x \in K \) . If \( Z = E \) we say that the space \( E \) is admissible. This notion plays an important role in the fixed point theory. The aim of this talk is to show the admissibility of some classes of Frechet spaces. As an application, it will be proved the admissibility of a large class modular spaces equipped with F-norms and norms which are certain generalizations of the classical Luxemburg F-norm and the classical Luxemburg and Orlicz norms in the convex case Also a linear version of admissibility (so-called metric approximation property) for order continuous symmetric spaces will be demonstrated. The talk is based on a joint work with Maciej Ciesielski, Instytut Matematyki Politechniki Poznańskiej.

Link to the talk.

Leave a Comment

Star-finite coverings of Banach spaces

In the first part of the talk, we provide a brief overview of tilings and coverings of Banach spaces. Then, some recent results in this context will be discussed. In particular, we will focus on star-finite coverings of Banach spaces by bodies. A family of subsets of a real normed space is called star-finite if each member of the family intersects at most finitely many other members. Part of the contents of the talk is included in joint work with C.A. De Bernardi, and L. Vesely.

Link to the talk.

Leave a Comment

Chalmers-Metcalf operator and minimal projections

The Chalmers-Metcalf operator is a powerful tool in the theory of minimal projections. It was introduced by two American mathematicians Bruce Chalmers and Frederic Metcalf and applied in finding a formula for minimal projection from \(C_R[a,b]\) onto the subspace polynomials of degree less than 2. During my talk, I would like to present various applications of this technique in the theory of minimal projections.

Link to the talk.

Leave a Comment