Funktsionaalanalüüsi seminar

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.).

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27.05.2021
kell 16.00

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 R^d. Some tools used in the proof, as well as related problems and applications, will be discussed.

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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.

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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:X→X with numerical radius 1 that almost attains its numerical radius at some state (x,x∗), there is a nearby compact operator S, also with numerical radius 1, that attains its numerical radius at a nearby state (y,y∗). 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 C0(L) 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.

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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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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]).

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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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 ℓ1 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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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 (ℓp,ℓq) 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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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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Diameter two properties in some vector-valued function spaces

In this talk, we introduce the space A(K,X) over a uniform algebra A on a compact Hausdorff space K. Every nonempty relatively weakly open subset of the unit ball of a vector-valued function space A(K,X) over an infinite dimensional uniform algebra has the diameter two. If X is uniformly convex, under the additional assumption of A⊗X⊂A(K,X), we show that Daugavet points and Δ-points on A(K,X) over 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,X) is also provided.

https://arxiv.org/abs/2103.04012
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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 k≥0 such that k∥T∥≤sup{|x∗(Tx)|:x∗∈X∗,x∈X,∥x∗∥=∥x∥=x∗(x)=1} for every T∈L(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 G∈L(X,Y) between two Banach spaces X and Y, the numerical index with respect to G, nG(X,Y), is the greatest constant k≥0 such that k∥T∥≤infδ>0sup{|y∗(Tx)|:y∗∈Y∗,x∈X,∥y∗∥=∥x∥=1,Rey∗(Gx)>1−δ} for every T∈L(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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(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 SX of every infinite-dimensional Banach space X 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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A characterization of the weak topology of the unit ball of purely atomic L1 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 L1-preduals.

A characterization of Banach spaces containingℓ1(κ) via ball-covering properties

Abstract. In 1989, G. Godefroy proved that a Banach space contains an isomorphic copy of ℓ1 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 ℓ1(κ+)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 αBX,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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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 T∈L(X) there is a modulus-one scalar θ such that

∥IdX+θT∥=1+∥T∥,

$$<!--//--><![CDATA[// ><!-- \| \operatorname{Id}_{X} + \theta T\| = 1 + \| T \|\,, //--><!]]>$$ 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 G∈L(X,Y) between two (real or complex) Banach spaces X,Y is said to be a if for every T∈L(X,Y) there exists a modulus-one scalar θ such that

∥G+θT∥=1+∥T∥.

$$<!--//--><![CDATA[// ><!-- \| G + \theta T\| = 1 + \| T\|\,. //--><!]]>$$ 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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$\mathrm{\Delta }$$\mathrm{}$$<!--//--><![CDATA[// ><!-- \Delta //--><!]]>$- 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 L1-spaces. We end with some open questions.

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Polynomial Daugavet Centers

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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 E=(Rn,∥⋅∥E)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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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.

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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