Functional analysis seminar | Institute of Mathematics and Statistics of University of Tartu

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  • View all other units

Contacts of UT units

Faculty of Arts and Humanities
Faculty phone: 
+372 737 5341
Faculty address: 
Jakobi 2, rooms 116–121, 51005 Tartu
  • Dean's Office
    Faculty phone: 
    + 372 737 5341
    Faculty address: 
    Jakobi 2, rooms 116–121, 51005 Tartu
  • Institute of History and Archaeology
    Faculty phone: 
    +372 737 5651
    Faculty address: 
    Jakobi 2, 51005 Tartu
  • Institute of Estonian and General Linguistics
    Faculty phone: 
    +372 737 5221
    Faculty address: 
    Jakobi 2, 51005 Tartu
  • Institute of Philosophy and Semiotics
    Faculty phone: 
    +372 737 5314
    Faculty address: 
    Jakobi 2, rooms 309–352, 51005 Tartu
  • Institute of Cultural Research
    Faculty phone: 
    (+372) 737 5223
    Faculty address: 
    Ülikooli 16, 51003 Tartu
  • School of Theology and Religious Studies
    Faculty phone: 
    +372 737 5301
    Faculty address: 
    Ülikooli 18–310, 50090 Tartu
  • College of Foreign Languages and Cultures
    Faculty address: 
    J. Liivi 4, 50409, Tartu
  • Viljandi Culture Academy
    Faculty phone: 
    +372 435 5232
    Faculty address: 
    Posti 1, 71004 Viljandi
Faculty of Social Sciences
Faculty phone: 
+372 737 5957
Faculty address: 
Lossi 36, 51003 Tartu
  • Dean's Office
    Faculty phone: 
    + 372 737 5900
    Faculty address: 
    Lossi 36, 51003 Tartu
  • Institute of Education
    Faculty phone: 
    +372 737 6440
    Faculty address: 
    Salme 1a–29, 50103 Tartu
  • Johan Skytte Institute of Political Studies
    Faculty phone: 
    +372 737 5582
    Faculty address: 
    Lossi 36–301, 51003 Tartu
  • School of Economics and Business Administration
    Faculty phone: 
    +372 737 6310
    Faculty address: 
    Narva mnt 18, 51009 Tartu
  • Institute of Psychology
    Faculty phone: 
    +372 737 5902
    Faculty address: 
    Näituse 2, 50409 Tartu
  • School of Law
    Faculty phone: 
    +372 737 5390
    Faculty address: 
    Näituse 20–324, 50409 Tartu
  • Institute of Social Studies
    Faculty phone: 
    +372 737 5188
    Faculty address: 
    Lossi 36, 51003 Tartu
  • Narva College
    Faculty phone: 
    +372 740 1900
    Faculty address: 
    Raekoja plats 2, 20307 Narva
  • Pärnu College
    Faculty phone: 
    +372 445 0520
    Faculty address: 
    Ringi 35, 80012 Pärnu
Faculty of Medicine
Faculty phone: 
+372 737 5326
Faculty address: 
Ravila 19, 50411 Tartu
  • Dean's Office
    Faculty phone: 
    +372 737 5326
    Faculty address: 
    Ravila 19, 50411 Tartu
  • Institute of Biomedicine and Translational Medicine
    Faculty phone: 
    +372 737 4210
    Faculty address: 
    Biomeedikum, Ravila 19, 50411 Tartu
  • Institute of Pharmacy
    Faculty phone: 
    +372 737 5286
    Faculty address: 
    Nooruse 1, 50411 Tartu
  • Institute of Dentistry
    Faculty phone: 
    +372 731 9856
    Faculty address: 
    Puusepa 1a, 50406 Tartu
  • Institute of Clinical Medicine
    Faculty phone: 
    +372 737 5323
    Faculty address: 
    L. Puusepa 8, 50406 Tartu
  • Institute of Family Medicine and Public Health
    Faculty phone: 
    +372 737 4190
    Faculty address: 
    Ravila 19, 50411 Tartu
  • Institute of Sport Sciences and Physiotherapy
    Faculty phone: 
    +372 737 5360
    Faculty address: 
    Ujula 4, 51008 Tartu
Faculty of Science and Technology
Faculty phone: 
+372 737 5820
Faculty address: 
Vanemuise 46–208, 51014 Tartu
  • Dean's Office
    Faculty phone: 
    +372 737 5820
    Faculty address: 
    Vanemuise 46–208, 51003 Tartu
  • Institute of Computer Science
    Faculty phone: 
    +372 737 5445
    Faculty address: 
    Narva mnt 18, 51009 Tartu
  • Estonian Marine Institute
    Faculty phone: 
    +372 671 8902
    Faculty address: 
    Mäealuse 14, 12618 Tallinn
  • Institute of Physics
    Faculty address: 
    W. Ostwaldi 1, 50411 Tartu
  • Institute of Chemistry
    Faculty phone: 
    +372 737 5261
    Faculty address: 
    Ravila 14a, 50411 Tartu
  • Institute of Genomics
    Faculty phone: 
    +372 737 4000
    Faculty address: 
    Riia 23b, 51010 Tartu
  • Institute of Mathematics and Statistics
    Faculty phone: 
    +372 737 5860
    Faculty address: 
    Narva mnt 18, 51009 Tartu
  • Institute of Molecular and Cell Biology
    Faculty phone: 
    +372 737 5027
    Faculty address: 
    Riia 23, 23b–134, 51010 Tartu
  • Tartu Observatory
    Faculty phone: 
    +372 737 4510
    Faculty address: 
    Observatooriumi 1, Tõravere, 61602 Tartumaa
  • Institute of Technology
    Faculty phone: 
    +372 737 4800
    Faculty address: 
    Nooruse 1, 50411 Tartu
  • Institute of Ecology and Earth Sciences
    Faculty phone: 
    +372 737 5835
    Faculty address: 
    Vanemuise 46, 51003 Tartu
Institutions
  • Library
    Faculty phone: 
    +372 737 5702
    Faculty address: 
    W. Struve 1, 50091 Tartu
  • Youth Academy
    Faculty phone: 
    +372 737 5581
    Faculty address: 
    Uppsala 10, 51003 Tartu
  • Museum
    Faculty phone: 
    +372 737 5674
    Faculty address: 
    Lossi 25, 51003 Tartu
  • University of Tartu Natural History Museum and Botanical Garden
    Faculty phone: 
    +372 737 6076
    Faculty address: 
    Vanemuise 46, 51003 Tartu
Support Units
  • Administrative Office
    Faculty phone: 
    +372 737 5606
    Faculty address: 
    Ülikooli 18a, 51005 Tartu
  • Centre for Entrepreneurship and Innovation
    Faculty phone: 
    +372 737 4809
    Faculty address: 
    Narva mnt 18, 51009, Tartu
  • University Office in Tallinn
    Faculty phone: 
    +372 737 6600
    Faculty address: 
    Teatri väljak 3, 10143 Tallinn
  • Estates Office
    Faculty phone: 
    +372 737 5137
    Faculty address: 
    Ülikooli 18a, 51005 Tartu
  • Finance Office
    Faculty phone: 
    +372 737 5125
    Faculty address: 
    Jakobi 4, 51005 Tartu
  • Grant Office
    Faculty phone: 
    +372 737 6215
    Faculty address: 
    Raekoja plats 9, 51004 Tartu
  • Information Technology Office
    Faculty phone: 
    +372 737 6000, IT-help: +372 737 5500
    Faculty address: 
    Ülikooli 18a, 51005 Tartu
  • Human Resources Office
    Faculty phone: 
    +372 737 5145
    Faculty address: 
    Ülikooli 18, rooms 302 and 304, 50090 Tartu
  • Internal Audit Office
    Faculty address: 
    Ülikooli 18-244, 51005 Tartu
  • International Cooperation and Protocol Office
    Faculty phone: 
    +372 737 6123
    Faculty address: 
    Ülikooli 18, rooms 104, 301, 305, 50090 Tartu
  • Marketing and Communication Office
    Faculty phone: 
    +372 737 5687
    Faculty address: 
    Ülikooli 18, rooms 102, 104, 209, 210, 50090 Tartu
  • Office of Academic Affairs
    Faculty phone: 
    +372 737 5620
    Faculty address: 
    Ülikooli 18, 50090 Tartu
  • Procurement Office
    Faculty phone: 
    +372 737 6632
    Faculty address: 
    Ülikooli 18a, 51005 Tartu
  • Rector's Strategy Office
    Faculty phone: 
    +372 737 5600
    Faculty address: 
    Ülikooli 18, 50090 Tartu
  • Student Council
    Faculty address: 
    Ülikooli 18b, 51005 Tartu
  • University of Tartu Press
    Faculty phone: 
    +372 737 5945
    Faculty address: 
    W. Struve 1, 50091 Tartu
Other Units
  • University of Tartu Academic Sports Club
    Faculty phone: 
    +372 737 5371
    Faculty address: 
    Ujula 4, 51008 Tartu
  • Tartu Student Village
    Faculty phone: 
    +372 740 9959
    Faculty address: 
    Narva mnt 25, 51013 Tartu
  • Tartu Students’ Club
    Faculty phone: 
    +372 730 2400
    Faculty address: 
    Kalevi 24, 51010 Tartu
  • Tartu University Hospital
    Faculty address: 
    L. Puusepa 1a, 50406 Tartu
  • University of Tartu Foundation
    Faculty phone: 
    +372 737 5852
    Faculty address: 
    Ülikooli 18, 50090 Tartu
  • View all other units

Functional analysis seminar

Narva mnt 18-2039 at 10:00-12:00 (GMT+3)

Join Zoom Meeting: https://ut-ee.zoom.us/j/95229664082?pwd=cmZNVEJSMFhkeWxlRXNEaVB3Umhpdz09
Meeting ID: 952 2966 4082
Passcode: 261772

When Who Topic
12.11.2021 Dmytro Seliutin (V. N. Karazin Kharkiv National University, Ukraine)  

04.11.2021

NB! Thursday is correct, at 10AM.

Jacopo Somaglia (University of Milan)

Norming M-bases in non-separable Banach spaces

In the first part of the talk we provide a brief overview on M-bases in non-separable Banach spaces, focusing on norming M-bases and some related classes of Banach spaces. Then, some recent results in this context will be discussed. In particular, a negative answer to the following longstanding open question will be presented: Must every Asplund Banach space with a norming M-basis be weakly compactly generated? Part of the contents of the talk are included in a recent joint work with P. Hájek, T. Russo, and S. Todorčević.

Video

29.10.2021

NB! The seminar starts at 5PM!

Javier Falcó Benavent (Universitat de València)

Norm attaining group invariant functionals

We will review some basic results of the norm attaining theory. We will also present some recent results in this theory in the case that the functionals are invariant under a group action on the space. The most basic examples of group invariant functionals are functionals on R2 that are invariant under permutations, that is, f(x,y)=f(y,x) for all x,y∈R. We will conclude this talk by presenting some group invariant results in the setting of norm attaining operators.

Video

22.10.2021

NB! The seminar begins at 11AM!

Ramón J. Aliaga (Universidad Politecnica de Valencia)

Extreme points of the unit ball in Lipschitz-free spaces

Lipschitz-free Banach spaces are currently the object of
intensive scrutiny, yet many of its basic geometric properties remain
shrouded in mystery. In this talk, we will discuss the problem of
determining the extreme points of the unit ball of the Lipschitz-free
space F(M) over a given metric space M. The problem, which is still
open in general, has been addressed by authors such as García-Lirola,
Pernecká, Petitjean, Procházka, Rueda Zoca, and Weaver. We will review
their contributions and go into some detail over the recent solution
for the case where M is compact (see https://arxiv.org/abs/2102.01219),
with focus on the main techniques for handling the general problem.

Slides

Video

15.10.2021 Geunsu Choi (Dongguk University, Seoul, South Korea)

The Birkhoff-James orthogonality and norm attainment for multilinear maps

Very recently, motivated by the result of Bhatia and Šemrl which characterizes the Birkhoff-James orthogonality of operators on a finite dimensional Hilbert space in terms of norm attaining points, the Bhatia-Šemrl property was introduced. In this talk, we investigate the denseness of the set of multilinear maps with the Bhatia-Šemrl property which is contained in the set of norm attaining ones. Contrary to the most of previous results which were shown for operators on real Banach spaces, we prove the denseness for multilinear maps on some complex Banach spaces. We also show that the denseness of operators does not hold when the domain space is c0 for arbitrary range. Moreover, we find plenty of Banach spaces Y such that only the zero operator has the Bhatia-Šemrl property in the space of operators from c0 to Y.

Slides

Video

8.10.2021 Alberto Salguero Alarcón (Universidad de Extremadura, Badajoz, Spain)

Classification of twisted sums of c0(I)

A twisted sum of two Banach spaces X and Y is another space Z containing Y as a closed subspace so that Z/Y = X. In this talk we focus on twisted sums of spaces of continuos functions on compact spaces. We first display some classical examples, and then we move to study the also standard spaces C(KA), where KA is the compact space generated by an almost-disjoint family A of subsets of N. They are twisted sums of c0 and c0(I), and this fact will prove essential in answering the following question: if C(KA) is isomorphic to C(KB), then... what?

Video

01.10.2021 Stefano Ciaci (University of Tartu)

Attaining strong diameter two property for infinite cardinals

A Banach space is said to have the strong diameter 2 property (SD2P) if every finite convex combination of slices of the unit ball has diameter 2 and it is said to have the attaining SD2P (ASD2P) if the diameter is attained by two elements of the unit sphere. What happens if we consider convex series of slices instead?

In order to answer this question, we extend the properties SD2P and ASD2P to infinite cardinals and we establish their dual connections with octahedral norms and Banach spaces failing the (−1)-ball-covering property. We also provide examples and, in particular, we characterize C(K) spaces and L1(μ) spaces satisfying the countable analogue of the ASD2P.

Slides

Video

03.06.2021 Triinu Veeorg (University of Tartu)

Daugavet- and Delta-points in Lipschitz-free spaces

In this talk we provide two characterizations for Daugavet-points in Lipschitz-free spaces that work for any metric space M. Furthermore, we apply this result to construct an example of a metric space M such that the corresponding Lipschitz-free space F(M) has the Radon–Nikodym property and also contains a Daugavet-point. As a last part of this talk we provide a characterization for Δ-points among convex combinations of molecules. Our main results generalize the ones from a recent preprint by Mingu Jung and Abraham Rueda Zoca (Daugavet points and Δ-points in Lipschitz-free spaces, to appear in Studia Math.).

Slides
Video

27.05.2021
at 16:00

Pedro L. Kaufmann (Federal University of São Paulo, Brazil)

Lipschitz-free spaces over compact Lipschitz manifolds

We will show that the Lipschitz-free space over a compact Lipschitz manifold of dimension d is isomorphic to the Lipschitz-free space over . Some tools used in the proof, as well as related problems and applications, will be discussed.

Slides
Video

20.05.2021 Andre Ostrak (University of Tartu)

The weak-star symmetric strong diameter 2 property in Lipschitz spaces

It is known that a Banach space is octahedral if and only if its dual has the weak-star strong diameter 2 property (w*-SD2P). Furthermore, in [PR] it was shown that a Lipschitz-free space is octahedral if and only if the corresponding metric space has the long trapezoid property. In this talk, we focus on the main results of two recent papers [O1], [O2], that provided similar characterisations to the weak-star symmetric strong diameter 2 property (w*-SSD2P) in Lipschitz spaces. In particular, we show that the w*-SSD2P is strictly stronger that the w*-SD2P in Lipschitz spaces, and we give a characterisation of the w*-SD2P of the Lipshitz space Lip₀(M) in terms of the corresponding metric space M and the corresponding Lipschitz-free space F(M).

References
[O1] A. Ostrak. Characterisation of the weak-star symmetric strong diameter 2 property in Lipschitz spaces. J. Math. Anal. Appl. 483 (2020), no. 2, 123630.
[O2] A. Ostrak. On the duality of the symmetric strong diameter 2 property in Lipschitz spaces. RACSAM 115, 78 (2021).
[PR] A. Prochazka, A. Rueda Zoca. A characterisation of octahedrality in Lipschitz-free spaces. Annales de l’Institut Fourier, 68 (2018) no. 2, 569-588.

Abstract
Slides
Video

13.05.2021 Óscar Roldán Blay (University of Valencia, Spain)

The Bishop-Phelps-Bollob'as property for numerical radius and compact operators

In this talk, we will present the Bishop-Phelps-Bollob'as property for numerical radius restricted to the case of compact operators (abbreviated BPBp-nu for compact operators). Roughly speaking, a Banach space X has this property if, whenever we have a compact operator T:Xwith numerical radius 1 that almost attains its numerical radius at some state there is a nearby compact operator S, also with numerical radius 1, that attains its numerical radius at a nearby state  We will first provide the necessary background to define this property and an initial wide list of Banach spaces satisfying the property. We will also see a tool that allows us to carry the property from some spaces to others, and we will use it to get more examples of spaces satisfying the property. Finally, we will see a visualization of the proof that all spaces have the property, whenever L is locally compact and Hausdorff. The contents of this talk are from a recent joint work with Domingo García, Manuel Maestre and Miguel Martín [GMMR2021].

[GMMR2021] D. García, M. Maestre, M. Martín, Ó. Roldán, On the compact operators case of the Bishop-Phelps-Bollob\'as property for numerical radius. To appear in Results in Mathematics.

Video
Slides

06.05.2021 Mingu Jung (Pohang University of Science and Technology, Republic of Korea)

When do all bounded linear operators attain their norm?

In this talk, we provide necessary and sufficient conditions for the existence of non-norm-attaining operators in L(E,F). By using a theorem due to Pfitzner on James boundaries, we show that if there exists a relatively compact set K of L(E,F) (in the weak operator topology) such that 0 is an element of its closure (in the weak operator topology) but it is not in its norm closed convex hull, then we can guarantee the existence of an operator which does not attain its norm. This allows us to provide the following generalization of results due to Holub and Mujica. We also present a characterization of the Schur property in terms of norm-attaining operators.

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29.04.2021 Katriin Pirk (University of Tartu, Estonia)

Stability results of diametral diameter two properties

The diametral diameter two properties were introduced being inspired by two settings. On the one hand, there were the diameter two properties (see e.g. [ALN]) where specific subsets of the unit ball (slices, relatively weakly open subsets, convex combinations of slices) have diameter 2. On the other hand, there was the research on the spaces with bad projections (see [IK]), inspired by the well-known Daugavet property. The spaces with bad projections admit to the following characterisation: every slice S of the unit ball contains an almost diametral point for each norm-1 element in S. Obviously, that slice S has diameter 2.

Connecting these two ideas a system of diametral diameter two properties similar to the three-property-system of the diameter two properties was started in [BLR]. The afore-mentioned characterisation of the spaces with bad projections became the slice version and two other definitions were added (for resp. relatively weakly open subsets and convex combinations of relatively weakly open subsets). In [AHLP] an additional diametral diameter two property, called the convex diametral diameter two property, was introduced. These notions have been researched over the years by many mathematicians (see e.g., [AHLP], [AHNTT], [BLR], [HPP] etc.).

In this seminary talk we give an overview of the diametral diameter two properties with the focus on the recent stability results regarding these properties obtained in a joint work with Johann Langemets (see [LP]).

The full text of [LP] is available at:
https://arxiv.org/pdf/2012.09492.pdf

References:
[AHLP] T. A. Abrahamsen, R. Haller, V. Lima, and K. Pirk, Delta- and Daugavet-points in Banach spaces, Proc. Edinb. Math. Soc. 63 (2020).

[AHNTT] T. A. Abrahamsen, P. Hájek, O. Nygaard, J. Talponen, and S. Troyanski, Diameter 2 properties and convexity, Studia Math. 232 (2016).

[ALN] T. A. Abrahamsen, V. Lima, and O. Nygaard, Remarks on diameter 2 properties, J. Convex Anal. 20 (2013).

[BLR] J. Becerra Guerrero, G. López-Pérez, and A. Rueda Zoca, Diametral Diameter Two Properties in Banach Spaces, J. Convex Anal. 25 (2018).

[HPP] R. Haller, K. Pirk, and M. Põldvere, Diametral strong diameter two property of Banach spaces is stable under direct sums with 1-norm, Acta Comment. Univ. Tartu Math. 20 (2016).

[IK] Y. Ivakhno and V. M. Kadets, Unconditional sums of spaces with bad projections, Visn. Khark. Univ., Ser. Mat. Prykl. Mat. Mekh. 645 (2004).

[LP] J. Langemets and K. Pirk, Stability results of diametral diameter two properties, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 115 (2021).

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22.04.2021
at 11:00
Miguel Martín (University of Granada, Spain)

Slicely countably determined sets, spaces, and operators

Slicely countably determined spaces (SCD spaces) were introduced by Aviles, Kadets, Martin, Meri, and Shepelska (Transactions of the American Mathematical Society 2010) as a topological property for separable Banach spaces which is satisfied by both Asplund spaces and spaces with the RNP (actually, by both spaces not containing  and strongly regular spaces). The property is defined in terms of a property of the slices of convex bounded subsets and produce interesting classes of bounded linear operators (SCD and HSCD operators). These ideas have been successfully applied to get important consequences in the theory of numerical index one spaces, for the Daugavet property, and for the study of spear operators between Banach spaces.

The objective of this seminar is to present the main examples and applications of the theory, comment some recent developments and present some open problems.

Some related references are the following ones:

  • Aviles, Martin, Meri, Shepelska. Slicely countably determined Banach spaces. Trans. Amer. Math. Soc. 362 (2010), 4871–4900.
  • Kadets, Martin, Meri, Perez. Spear operators between Banach spaces. Lecture Notes in Math. 2205, 2018.
  • Kadets, Perez, Werner. Operations with slicely countably determined sets, Funct. Approx. 59 (2018), 77–98.

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15.04.2021
at 12:00

Luis C. García Lirola (University of Zaragoza, Spain)

The Weak Maximizing Property and asymptotic geometry of Banach spaces

Recently, a new property related to norm-attaining operators between Banach spaces has been introduced: the weak maximizing property (WMP). In this talk, we provide new sufficient conditions, based on the moduli of asymptotic uniform smoothness and convexity, which imply that a pair (X,Y) enjoys the WMP. This approach not only allows us to (re)obtain as a direct consequence that the pair has the WMP, but also provides many more natural examples of pairs having a given maximizing property. This is part of a joint work with Colin Petitjean.

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08.04.2021 Nikita Leo (University of Tartu, Estonia)

Plasticity of the unit ball of c0

In this talk we consider a challenging open problem of whether the unit ball of every Banach space is a plastic metric space (a metric space is called plastic if every non-expansive bijection from this space onto itself is an isometry). We are going to provide some insight into the problem and show what has been achieved so far. We also present a new result, which says that the unit ball of c0 has a property similar to plasticity - we show that a non-expansive bijection from the unit ball of c0 onto itself is an isometry, provided that the inverse map is continuous.

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01.04.2021 Hyung-Joon Tag (Dongguk University, Republic of Korea)

Diameter two properties in some vector-valued function spaces

In this talk, we introduce the space A(K,Xover a uniform algebra on a compact Hausdorff space K. Every nonempty relatively weakly open subset of the unit ball of a vector-valued function space A(K,Xover an infinite dimensional uniform algebra has the diameter two. If X is uniformly convex, under the additional assumption of AXA(K,X), we show that Daugavet points and Δ-points on A(K,Xover a uniform algebra A are the same, and they are characterized by the norm-attainment at a limit point of the Shilov boundary of A. In addition, a sufficient condition for the convex diametral local diameter two property of A(K,Xis also provided.

https://arxiv.org/abs/2103.04012
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25.03.2021 Alicia Quero de la Rosa
(University of Granada, Spain)

Numerical index with respect to an operator

The concept of numerical index was introduced by Lumer in 1968 in the context of the study and classification of operator algebras. This is a constant relating the norm and the numerical range of bounded linear operators on the space. More precisely, the numerical index of a Banach space X, n(X), is the greatest constant k0 such that kTsup{|x(Tx)|:xX,xX,x=x=x(x)=1for every TL(X).

Recently, Ardalani introduced new concepts of numerical range, numerical radius, and numerical index, which generalize in a natural way the classical ones and allow to extend the setting to the context of operators between possibly different Banach spaces. Given a norm-one operator GL(X,Y) between two Banach spaces X and Y, the numerical index with respect to GnG(X,Y), is the greatest constant ksuch that kTinfδ>0sup{|y(Tx)|:yY,xX,y=x=1,Rey(Gx)>1δfor every TL(X,Y).

In this talk, we will give an overview of the topic, analysing differences and similarities between these concepts, and presenting some classical and recent results in the area.

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18.03.2021 Tomasso Russo (Czech Academy of Sciences, Czech Technical University in Prague)

(1+)-meters apart: Separated sets in Covid times

Riesz’ lemma, one of the most classical results in Functional Analysis and now proved in the first lectures of every course in the topic, asserts that the unit sphere of every infinite-dimensional Banach space contains a sequence of points whose mutual distances are at least 1, thereby demonstrating the non-compactness of the unit ball in infinite dimensions. The, by now rather wide, topic of separated sets in Banach spaces can be safely considered to originate from such a lemma.

In our talk, we will survey some classical and recent results in the area, also pointing out some problems that remain open. We shall discuss the main ideas behind the proofs of selected results, as an illustration of some techniques in the area (both in the separable and the non-separable contexts). The talk is intended to be elementary and students-friendly, the unique real prerequisites being the definition of a Banach space and the Hahn–Banach theorem.

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11.03.2021 Rubén Medina (University of Granada, Spain; Czech Technical University in Prague, Czech Republic)

A characterization of the weak topology of the unit ball of purely atomic preduals

We study Banach spaces with a weak stable unit ball, that is, Banach spaces where every convex combination of relatively weakly open subsets of the unit ball is again relatively weakly open in its unit ball. A walkthrough of previous overall results will be done, as well as a more concise dedication to the newer results. The main result of the talk will be a characterization of the spaces with weak stable unit ball among the -preduals.

https://arxiv.org/abs/2012.04940
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04.03.2021 Stefano Ciaci (University of Tartu, Estonia)

A characterization of Banach spaces containing via ball-covering properties

Abstract. In 1989, G. Godefroy proved that a Banach space contains an isomorphic copy of  if and only if it can be equivalently renormed to be octahedral. It is known that octahedral norms can be characterized by means of covering the unit sphere by a finite number of balls. This observation allows us to connect the theory of octahedral norms with ball-covering properties of Banach spaces introduced by L. Cheng in 2006. Following this idea, we extend G. Godefroy’s result to higher cardinalities. We prove that, for an infinite cardinal κ, a Banach space X contains an isomorphic copy of if and only if it can be equivalently renormed in such a way that its unit sphere cannot be covered by κ many open balls not containing where α(0,1). We also investigate the relation between ball-coverings of the unit sphere and octahedral norms in the setting of higher cardinalities.

https://arxiv.org/abs/2102.13525
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25.02.2021 Antonio Pérez Hernández (UNED, Spain)

Spear operators between Banach spaces

Abstract. The concept of spear operator is intimately related to the notions of numerical range, numerical radius and numerical index introduced for Banach spaces by Lumer and Bauer (1960s). Duncan et al. (1970s) showed that Banach spaces X having are characterized as those satisfying that for every TL(X) there is a modulus-one scalar θ such that

IdX+θT=1+T,

also providing the first examples and properties. The geometrical properties of these spaces have been extensively studied, searching for characterizations that do not involve operators, and leading to the introduction of new properties (lushness, alternative Daugavet property…) that have become useful in other settings.

More recently, Ardalani (2014) presented more general definitions of numerical range, numerical radius and numerical index in the context of operators between possibly different Banach spaces. In particular, he introduced the concept of and initiated the study of general spear operators: a norm-one operator GL(X,Y) between two (real or complex) Banach spaces X,Y is said to be a if for every TL(X,Y) there exists a modulus-one scalar θ such that

G+θT=1+T.

In this talk, we will give an overview of the topic, reviewing main results and ideas, and posing some open problems on the way.

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04.12.2020 Abraham Rueda Zoca (University of Murcia, Spain)

Δ- and Daugavet points in Lipschitz-free spaces

Abstract. In this talk we will focus on giving necessary and sufficient conditions for a Lipschitz-free spaces F(M) to have Δ- and Daugavet points. As a consequence of our study, we will provide examples of metric spaces M and molecules in F(M) which are Δ-points but not Daugavet points, which is a completely different behaviour to the case of -spaces. We end with some open questions.

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27.11.2020 Elisa Regina dos Santos (Federal University of Uberlândia, Brazil)

Polynomial Daugavet Centers

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20.11.2020 Olesia Zavarzina (V.N. Karazin Kharkiv National University, Ukraine)

Generalized-lush spaces and connected problems

Abstract. The talk is devoted to geometric properties of GL-spaces. We will demonstrate that every finite-dimensional GL-space is polyhedral; We also characterise the spaces with absolute norm such that for every finite collection of GL-spaces their E-sum is a GL-space (GL-respecting spaces). We will also give the classification of GL- and GLR-spaces in dimention 2.

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13.11.2020

Sheldon Dantas (Czech Technical University in Prague, Czech Republic)

Gonzalo Martínez Cervantes (University of Murcia, Spain)

Octahedral norms in Free Banach Lattices

Abstract. Our aim in this talk is twofold. First of all, we present an overview on basic facts about free Banach lattices generated by a given Banach space E, which will be denoted by FBL[E]. We present some properties and known results about such a space. This will give all the necessary background we need for the second part of the talk, which consists on the study of octahedral norms in FBL[E]. We provide some sufficient conditions so that such a space has an octahedral norm. We also discuss almost squareness and Fréchet differentiability on these spaces. We conclude the talk by presenting some natural questions related to the Daugavet property.

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23.10.2020 André Martiny (University of Agder, Norway)

Daugavet- and delta-points in Banach spaces with unconditional bases

Abstract. We study the existence of Daugavet- and delta-points in the unit sphere of Banach spaces with a 1-unconditional basis. A norm one element x in a Banach space is a Daugavet-point (resp. delta-point) if every element in the unit ball (resp. x itself) is in the closed convex hull of unit ball elements that are almost at distance 2 from x. A Banach space has the Daugavet property (resp. diametral local diameter two property) if and only if every norm one element is a Daugavet-point (resp. delta-point). It is well-known that a Banach space with the Daugavet property does not have an unconditional basis. Similarly spaces with the diametral local diameter two property do not have an unconditional basis with suppression unconditional constant strictly less than 2.

We show that no Banach space with a subsymmetric basis can have delta-points. In contrast we construct a Banach space with a 1-unconditional basis with delta-points, but with no Daugavet-points, and a Banach space with a 1-unconditional basis with a unit ball in which the Daugavet-points are weakly dense.

https://arxiv.org/abs/2007.04946
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