
Jaime Castillo  Composition operators of spaces of bounded Dirichlet series
Universitat de València
+ 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.

Rubén Medina Sabino  Ahondando en el fenómeno de slices grandes
Universidad de Granada
+ info In contrast with one of the wellknown geometric characterizations of the RadonNikodym 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 vectorvalued maps. 
Alicia Quero de la Rosa  Índice numérico respecto a un operador
Universidad de Granada
+ 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 normone 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.

Sergi Baena Miret  No hay espacios $\Lambda^p_u(w)$ solo espacios $L^2$ con pesos
Universidad de Barcelona
+ info An important property of the $A_p$ weights is the extrapolation theorem of Rubio de Francia. It was announced in 1982^{1} and given with a detailed proof in 1984^{2}, 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 Córdoba^{3} 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 $\Lambda^p_u(w)$.
 J.L Rubio de Francia: Factorization and extrapolation of weights. Bull. Amer. Math. Soc. 7 (1982), 393–395.
 J.L Rubio de Francia: Factorization theory and $A_p$ weights. Amer. J. Math. 106 (1984), no. 3, 533–547.
 J. GarcíaCuerva: José Luis Rubio de Francia (1949–1988). Collect. Math.38 (1987), no. 1, 3–15.

Abraham Rueda Zoca  Locally complemented subspaces and free objects
Universidad de Murcia
+ 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 Lipschitzfree spaces and to free Banach lattices.
The content of this talk is part of a work in progress in collaboration with Antonio Avilés, Gonzalo MartínezCervantes and José David RodríguezAbellán.
The author was supported by Juan de la CiervaFormación fellowship FJC2019039973, by MTM201786182P (Government of Spain, AEI/FEDER, EU), by MICINN (Spain) Grant PGC2018093794BI00 (MCIU, AEI, FEDER, UE), by Fundación Séneca, ACyT Región de Murcia grant 20797/PI/18, by Junta de Andalucía Grant AFQM484UGR18 and by Junta de Andalucía Grant FQM0185.

Francisco Javier González Doña  Invariant Subspaces For RankOne Perturbations Of Diagonal Operators
ICMAT y Universidad Complutense de Madrid
+ info We will address results concerning the existence of nontrivial reducing subspaces for rankone perturbations of diagonal operators on separable, complex Hilbert spaces, showing that the spectral picture is essential at this regard. Likewise, we will present a characterization of spectral subspaces for such operators, which will allow us to enlarge the subclass of rankone perturbations of diagonal operators which are known to have nontrivial invariant subspaces. (Joint works with Eva A. GallardoGutiérrez) 
Josefa Caballero Mena  Existence and uniqueness of positive solutions to a class of singular integral boundary value problems of fractional order with parametric dependence
Universidad de Las Palmas de Gran Canaria
+ info In this paper, we study the existence and uniqueness of positive solutions for the following singular boundary value problem$$\left\{\begin{eqnarray*} && D^{\alpha}_{0^+} u(t)+f(t,u(t),(Hu)(t))=0, \ \ 0<t<1,\\ && u(0)=u'(0)=\ldots = u^{(n2)}(0)=0,\\ &&u(1)=\lambda \int_0^1 u(s)ds,\end{eqnarray*}\right.$$
where $n\geq 3$, $n1<\alpha\leq n$, $\lambda \in (0,\alpha)$, $H$ is an operator applying $C[0,1]$ into itself and $f:(0,1]\times [0,\infty)\times [0,\infty)\to [0,\infty)$ ($f$ can have a singularity at $t_0=0$) and $D^{\alpha}_{0^+}$ denotes the RiemannLiouville fractional derivative. The main tool used in the proof of the results is a recent fixed point theorem in complete metric spaces.
Joint work with J. Harjani, K. Sadarangani

Antoni López Martínez  Frequently recurrent operators and stronger recurrent notions
Universidad Politécnica de Valencia
+ info Motivated by some results of Costakis et al. on the notion of recurrence in linear dynamics, we will present various stronger forms of recurrence for linear operators, in particular that of frequent recurrence. We will also expose the relationship between each type of recurrence and the corresponding notion of hypercyclicity, paying attention to the size of the set of frequently recurrent vectors. We finally will introduce the notion of Frecurrence, being F a Furstenberg family of integers, which results on stronger forms of recurrence such as uniform recurrence. A. Bonilla, K.G. GrosseErdmann, A. LópezMartínez and A. Peris: Frequently recurrent operators. arXiv:2006.11428, 2020.
 G. Costakis, A. Manoussos and I. Parissis: Recurrent linear operators. Complex Anal. Oper. Theory 8 (2014), 1601–1643.
 V. J. Galan, F. MartínezGimenez, P. Oprocha and A. Peris, Product recurrence for weighted backward shifts, Appl. Math. Inf. Sci. 9 (2015), 23612365.
 S. Grivaux, E. Matheron and Q. Menet, Linear dynamical systems on Hilbert spaces: Typical properties and explicit examples, Mem. Amer. Math. Soc., to appear.

Sheldon Miriel Gil Dantas  A characterization of a local vector valued Bollobás theorem
Universitat Jaume I
+ info In this (friendlylike) talk, we are interested in giving a characterization for a local vector valued Bollobás type theorem. This generalizes at once some results due to Chakraborty, D. Sain, and J. Talponen. Moreover, we present a complete characterization for when this Bollobás type theorem is satisfied for functionals on projective tensor products between Banach spaces under strict convexity or KadecKlee property assumptions. As a consequence, we generalize some results in the literature related to the strong subdifferentiability of the projective tensor product. This is a joint work with Abraham Rueda Zoca. 
Gonzalo Martínez Cervantes  Classical and new results in Banach Lattice Theory
Universidad de Murcia
+ info A Banach latice is a Banach space equipped with compatible lattice operations. Most classical Banach spaces are Banach lattices. Indeed, during the first half of the 20th century the importance of studying Banach lattices in connection with Banach Space Theory was widely recognized. Nevertheless, according to H.H. Schaefer, after that period Banach Lattice Theory could not quite keep pace with the rapid development of general functional analysis and thus developed into a theory largely existing for its own sake.
Understanding the relation between the linear and lattice structures of Banach lattices has been the driving force behind a large part of research in this field. For instance, a wellknown classical result states that a Banach lattice contains a subspace isomorphic to $c_0$ if and only if it contains a sublattice isomorphic to $c_0$. The problem we address in this talk is to characterize those Banach lattices which share this property with $c_0$. Improving results of H.P. Lotz, H.P. Rosenthal and N. Ghoussoub, we will show that a Banach lattice contains a subspace isomorphic to $C[0,1]$ if and only if it contains a sublattice isomorphic to $C[0,1]$.
This talk is based on a recent joint work with Antonio Avilés, Abraham Rueda Zoca and Pedro Tradacete.