For 1<p≤∞, we study the Borel complexity, in the class of all separable Banach spaces, and the existence of universal spaces for three classes of separable Banach spaces, denoted Tp, Ap and Np. These classes are defined in terms of the existence of upper p-estimates for weakly null trees. The first one, Tp, is known to coincide with the class of Banach spaces admitting an equivalent norm which is asymptotically uniformly smooth with power type p. In this talk we will focus on the following questions: what is the topological complexity of these classes: Borel or analytic non Borel? Do they contain a space that is both injectively and surjectively universal for the class? If not, can we  build  small families of Banach spaces that are both injectively and surjectively universal for the class?

We give an overview of the most important classical results on norm attaining linear operators, in most cases explaining the recent advances and the possible open problems with clues (if we know any!). The list of results include, at least, the following:

- Necessary conditions on the domain space to get that all operators can be approximated by norm attaining operators (Lindenstrauss 1963, Bourgain 1978, and recent results from [2]).

- The relation with the Radon-Nikodým property, including the quasi norm attaining version to deal with range spaces (Bourgain 1978, Huff 1980, and recent results from [1]).

- Compact operators and finite rank operators (Lindenstrauss 1963, Johnson-Wolfe 1979, Schachermayer 1983, Martín 2014, recent results from [2] and [3], and a discussion on how to deal with the open problems).

In this talk there will not be other type of functions than bounded linear operators, nor other related properties like Bollobás' type results. 

References:

[1] Choi, Choi, Jung, Martin, On quasi norm attaining operators between Banach spaces, RACSAM 116 (2022), 133

[2] Jung, Martin, Rueda Zoca, Residuality in the set of norm attaining operators between Banach spaces, J. Funct. Anal. 284 (2023), 109746.

[3] Kadets, López, Martín, Werner, Norm attaining operators of finite rank, In: The Mathematical Legacy of Victor Lomonosov, pp. 157-183. Advances in Analysis and Geometry, 2. De Gruyter, Berlin, 2020.

We shall report on a joint project with Sheldon Dantas and Petr Hájek concerning smooth renormings of (not necessarily complete) normed spaces. It is a well-known fact that the existence of a C¹-smooth norm on a separable Banach space X implies the separability of the dual X*. It follows, for example, that no infinite-dimensional closed subspace of l1 can admit a differentiable norm. However, the situation is radically different if we only consider (not necessarily closed) linear subspaces; for example, there is an infinite-dimensional linear subspace of l1 that admits a C∞-smooth norm. Indeed, one of the first results in this area, due to Hájek in 1995, asserts that every separable Banach space admits a dense subspace with a C∞-smooth norm. This led us to consider the following question:

Let X be a Banach space. Does there exist a dense subspace Y of X 

that admits a Cᵏ-smooth norm, for some k?

In the talk we shall review the status of the art on the said question and we shall touch upon more recent results. Along the way, we will also mention some open problems and possible directions for further research.

We analyze question 18 of J. Lindenstrauss in [4]. We prove that  a Banach space E with a norming subspace F ⊆ E* has an equivalent σ(E,F)-lower semicontinuous LUR norm if,  and only if,  there is a sequence {An: n=1,2,...} of subsets of E such that, given any x∈E and ε>0, there is a σ(E,F)-open half-space H and p ∈ IN such that x∈ H ∩ Ap and the slice H ∩ Ap can be covered with  countable many sets of diameter less than ε. Thus E has an equivalent σ(E,F)-lower semicontinuous LUR norm if, and only if,  it has another one with separable denting faces [8,9].

This result completely solves four problems asked in [6, Question 6.33, p.128] extending Troyanski's fundamental results (see Chapter IV in [1]), and other ones in [2,5]. 

Moreover,  LUR renormings are possible at points of separable faces wich could be glued as a σ-slicely isolated family of faces [6],  of the unit sphere  of E.

Among new examples covered by this results are Banach spaces C(K), where K is  a Rosenthal compact space K ⊆ IR^Γ i.e., a compact space of Baire one functions on a Polish space Γ,  with at most  countably many discontinuity points for every $s ∈ K$,  which solves three problems asked in [6, Question 6.23,  p.125]. Previously,  it was only known for K being separable too,  see [3] where the σ-fragmentability of C(K) was already proved for non separable K, and a conjecture for the pointwise lower semicontinuous and  LUR renorming presented here was posed, details  will appear in [7].

For strictly convex  renormings, we solve a recent question of R. Smith [11] giving a final answer to Lindenstauss question 18 in [4], see [9] and [10]. Indeed, we prove that E admits an equivalent σ(E,F)-lower semicontinuous and strictly convex norm if, and only if, it has another one with separable faces. A purely topological new characterization follows for dual spaces and dual norms.

References

[1] R. Deville, G. Godefroy, and V. Zizler. Smoothness and renormings in Banach spaces, volume 64 of Pitman Monographs and Surveys in Pure and Applied Mathematics. Longman Scientific & Technical, Harlow, 1993.

[2] F. García, L. Oncina, J. Orihuela, and S. Troyanski. Kuratowski's index of non-compactness and renorming in Banach spaces. J. Convex Anal., 11(2):477-494, 2004.

[3] R. Haydon, A. Moltó, and J. Orihuela. Spaces of functions with countably many discontinuities. Israel J. Math., 158:19-39, 2007.

[4] J. Lindenstrauss. Some open problems in Banach space theory.

In Seminaire Choquet. Initiation  a l' analyse tome 15, 1975-1976, Exp. No. 18, pages 1{9. Secretariat mathématique, Paris, 1975-76.

[5] A. Moltó, J. Orihuela, and S. Troyanski. Locally uniformly rotund renorming and fragmentability. Proc. London Math. Soc.

(3), 75(3):619-640, 1997.

[6] A. Moltó J. Orihuela, S. Troyanski, and M. Valdivia. A non-linear transfer technique for renorming, volume 1951 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2009.

[7] A. Moltó, V. Montesinos, J. Orihuela and S. Troyanski. Rosenthal compact spaces and renormings. Work in progress.

[8] V. Montesinos and J. Orihuela. Separable slicing and locally uniformly rotund renormings of Banach spaces. To appear in PAFA, 2024.

[9] V. Montesinos and J. Orihuela. Weak compactness and separability of faces for convex renormings of Banach spaces. Preprint 2024.

[10] J. Orihuela. A separation method through convex sets for strictly convex renormings of Banach spaces. Work in progress.

[11] R.J. Smith. A topological characterization of dual strict convexity in Asplund spaces. J. Math. Anal. Appl. 485 (2020) 123806

To be announced.

We survey the known constructions of generic (non-expansive, universal, approximately homogeneous) operators on separable Banach spaces. An operator can be called generic if it is non-expansive and the family of its isometric copies forms a residual set with respect to some "reasonable" operator topology. We address the question which classical Banach spaces admit a generic operator.

We show, using basic tools from category theory, that there exists a generic operator on the sequence space l1. We shall also discuss generic Banach spaces, where genericity is defined through a natural infinite game.

Part of the talk is based on a work (in progress) with J. Garbulinska-Wegrzyn and S. Turek.

We will discuss several notions of basis in Banach spaces and Banach lattices depending on the notion of convergence for the approximation of a vector by linear combination of the basis. We shall use descriptive set-theoretic tools for solving several problems from the literature. Joint work with Christian Rosendal, Mitchell Taylor and Pedro Tradacete.

We will discuss certain aspects of the topological dynamics of the group of linear isometries Iso(E) of a Banach space E, including  the envelopes associated to them, and "isotropic quotients" B_Eⁿ//Iso(E). This is a joint work (in progress) with V. Ferenczi.

While C(βω) is a classical example of a Grothendieck space, C(βω x βω) contains a complemented copy of c0 —this is a consequence of a general result due to Cembranos and Freniche. Alspach and Galego [1] asked if the space C(βω x βω) contains complemented copies of other separable infinite dimensional Banach spaces. We present a result implying that C(βω x βω) contains a complemented copy of C[0,1]. In fact, we prove that C[0,1] is isomorphic to a complemented subspace of any space of the form C(K x L) where K and L are non-scattered compacta. This is joint work with Jakub Rondoš and Damian Sobota.

References

[1] D.E. Alspach, E. M. Galego, Geometry of the Banach spaces C(βN x K,X), Studia Math., 2011, 207(2), 153–180.

We shall recall some results on complex structures in Banach spaces, then address the following question: if a C-linear operator on a complex space is strictly singular, then must it be strictly singular as an R-linear operator?

We shall answer the question positively, as part of a general criterion in the context of operator ideals in the sense of Pietsch.

A (separable) twisted Hilbert space X is a Banach space containing an isomorphic copy of l2 and such that the respective quotient is also isomorphic to l2. We represent that by the diagram

0 → l2 → X → l2 → 0

We present an ``infinite dimensional cone'' of twisted Hilbert spaces satisfying:

• The quotient map q is strictly singular, and the inclusion map i strictly cosingular.

• Each space is isomorphic to its dual, but not to its conjugate dual.

• Each space appears as the derived space at 0 of complex interpolation of a family of four Banach spaces.

This is a joint work with Sheldon Dantas (Universidad de Valencia) and Daniel L. Rodríguez-Vidanes (Universidad Politécnica de Madrid).

We will discuss some results on domination and factorization for certain classes of multilinear and Lipschitz operators. We will focus on certain variants of (q, p)-summability of these operators. In particular, we prove an integral domination theorem for Lipschitz operators from C(K)-spaces to metric spaces. As an application, we recover Pisier’s theorem on factorization through Lorentz spaces Lq,1(μ) for (q, 1)-summing linear operators with 1 < q < ∞. We will also present a theorem that characterizes those Lipschitz operators from any quasi-metric space to L0(μ) that factorize through weighted Marcinkiewicz spaces. We apply this result to sublinear operators mapping quasi-Banach spaces with generalized Rademacher type generated by Orlicz sequence spaces. As a consequence, we obtain the classical Nikishin theorem. The talk is based on joint works with Enrique A. Sánchez Pérez.