+ info The theory of Dirichlet series has played an important role in analytic number theory and in complex analysis, but also, they show very interesting links with the theory of spaces of functions in infinitely many variables in functional analysis. This talk starts with an approachable introduction to Dirichlet series and multiple Dirichlet series, to later showcase their versatility as they link to so many fields in mathematics. The final aim will be to explore the problem of composition operators of Dirichlet series, both in the simple and in the multiple case.

The composition operators of spaces of Dirichlet series were studied first by Gordon and Hedenmalm in 1999. In their first paper about this subject, the authors characterized the form of the symbol of a composition operator acting on any space of Dirichlet series containing the Dirichlet monomials. In this talk we explore the main reasons for which a symbol of a composition operator acting on spaces of Dirichlet series is characterized in the way it is, and we try to build the analogue of those symbols for the case of double Dirichlet series, extending the same reasoning the double case.

+ info In contrast with one of the well-known geometric characterizations of the Radon-Nikodym property of Banach spaces, in this talk we are going to focus on spaces whose unit ball has slices (or even convex combination of slices) of diameter 2. In fact, we are going to focus on a stronger property concerning the weak topology. We will say that a unit ball is weak stable whenever every convex combination of relatively weakly open subsets of that ball is relatively weakly open. This preperty is in strong connection with the big slices phenomenon and have a lot of other important consequences. We will see some of those consequences as well as the connection with diameter 2 properties. We will also give some new examples of spaces with weak stable unit balls inside the family of $L_1$-preduals and spaces of continuous vector-valued maps.

+ info 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 $k\geq 0$ such that

$k\|T\|\leq \sup\bigl\{|x^\ast(Tx)|\colon x^\ast\in X^\ast,\,x\in X,\,\|x^\ast\|=\|x\|=x^\ast(x)=1\bigr\}$

for every $T\in\mathcal{L}(X)$.

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 $G\in \mathcal{L}(X,Y)$ between two Banach spaces $X$ and $Y$, the numerical index with respect to $G$, $n_G(X,Y)$, is the greatest constant $k\geq 0$ such that

$ k\|T\|\leq \inf_{\delta>0} \sup\bigl\{|y^\ast(Tx)|\colon y^\ast\in Y^\ast,\,x\in X,\,\|y^\ast\|=\|x\|=1,\,\operatorname{Re} y^\ast(Gx)>1-\delta\bigr\} $

for every $T\in \mathcal{L}(X,Y)$.

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.

+ info An important property of the $A_p$ weights is the extrapolation theorem of Rubio de Francia. It was announced in 1982¹ and given with a detailed proof in 1984², both by J.L. Rubio de Francia. In its original version, reads as follows: if $T$ is a sublinear operator which satisfies the strong type boundedness

$T:L^2(v)\rightarrow L^2(v)$

for every weight $v \in A_2$ with constant only depending on $v$, then for $1 < p < \infty$,

$T:L^p(v)\rightarrow L^p(v)$

is bounded for every $v \in A_p$, with constant depending only on $v$. This leaded Antonio Córdoba³ to assert: “There are no $L^p$ spaces, only weighted $L^2$”. Indeed, this result can be extended, in particular, to the weighted Lorentz spaces $\Lambda^p_u(w)$.

1. J.L Rubio de Francia: Factorization and extrapolation of weights. Bull. Amer. Math. Soc. 7 (1982), 393–395.
2. J.L Rubio de Francia: Factorization theory and $A_p$ weights. Amer. J. Math. 106 (1984), no. 3, 533–547.
3. J. García-Cuerva: José Luis Rubio de Francia (1949–1988). Collect. Math.38 (1987), no. 1, 3–15.

+ info The concept of locally complemented Banach spaces was introduced by N. J. Kalton in 1984 and, since then, it has been intensively studied because of the good behaviour in connection with other properties of Banach spaces (for instance, it is known that locally complemented Banach spaces inherit the approximation property). In this talk, we will exhibit a characterisation of locally complemented subspaces in terms of a property of extension of bounded operators taking values in dual Banach spaces. We will apply this characterisation to analyse how this property is preserved by taking free objects, paying special attention to the case of Lipschitz-free spaces and to free Banach lattices.

The content of this talk is part of a work in progress in collaboration with Antonio Avilés, Gonzalo Martínez-Cervantes and José David Rodríguez-Abellán.

The author was supported by Juan de la Cierva-Formación fellowship FJC2019-039973, by MTM2017-86182-P (Government of Spain, AEI/FEDER, EU), by MICINN (Spain) Grant PGC2018-093794-B-I00 (MCIU, AEI, FEDER, UE), by Fundación Séneca, ACyT Región de Murcia grant 20797/PI/18, by Junta de Andalucía Grant A-FQM-484-UGR18 and by Junta de Andalucía Grant FQM-0185.