Madrid, 8–11 February 2016 • Facultad de Ciencias Matematicas • Universidad Complutense de Madrid • SPAIN

FUNCTION THEORY ONINFINITE DIMENSIONAL SPACES XIVOrganizing committee: J. Jaramillo, J. Lopez-Salazar, O. Madiedo, A. Prieto

BOOK OF

ABSTRACTS

FT DS2016

1

Index of Abstracts(In this index, in case of multiple authors only the speaker is shown)

Plenary TalksR. Aron Algebras of symmetric analytic functions (bis) . . . 7

G. Botelho Polynomial ideals from a nonlinear viewpoint . . . 7

K. C. Ciesielski Lineability and additivity cardinals for real-valuedfunctions: old results and new developments . . . . . 7

A. Daniilidis Self-contracted curves: from Euclidean spaces toRiemannian manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 8

R. Deville Embedding metric spaces into c0 . . . . . . . . . . . . . . . . 8

V. Dimant On the structure of Pw(nE,F ) as a subspace ofP(nE,F ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

E. Durand Preservation of Poincare inequalities under spher-icalization and flattening in the metric setting . . . 10

R. Espınola Continuous selections of Lipschitz extensions inBanach and metric spaces . . . . . . . . . . . . . . . . . . . . . . . 10

A. Ibort A Radon transform on infinite dimensions: thefate of Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

S. Lajara Strongly weakly compactly generated Banachspaces and their relatives . . . . . . . . . . . . . . . . . . . . . . . . 11

A. Lemenant Regularity of 1D almost minimal sets in Banachspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

M. Maestre The algebra of the ball . . . . . . . . . . . . . . . . . . . . . . . . . . 11

D. Puglisi Spaces of compact operators . . . . . . . . . . . . . . . . . . . . . 12

2

Short TalksD. Azagra Morse and Sard meet Taylor and Stepanov . . . . . . 15

M. Bienias Everywhere surjective and Sierpinski-Zygmundfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

M. Candan Optimal Control Problem For Processes Given ByMulti-Parameter Linear Stochastic Dynamic Sys-tem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

A. Dalet On Lipschitz-free spaces of ultrametric spaces . . . 16

M. Dymond Porosity of contractions in spaces of non-expansive mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

L. Garcıa-Lirola Asymptotic uniform smoothness in spaces of com-pact operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

R. Gonzalo Asymptotic smoothness, convex envelopes andpolynomial norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

A. J. Guirao Completeness of Mackey Topologies on BanachSpaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

J. Marchwicki On the set of limit points of conditionally conver-gent series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

A. Merono More on Lipschitz-type functions . . . . . . . . . . . . . . . . 20

A. Miralles Interpolating sequences for Bloch spaces on theunit ball of a Hilbert space . . . . . . . . . . . . . . . . . . . . . . 20

L. A. de Moraes Topological and algebraic properties of spaces ofLorch analytic mappings . . . . . . . . . . . . . . . . . . . . . . . . 21

A. Or Successive Approximation Method in MatrixGames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

V. Payne A Dissipative System Driven By A Minimal Tur-bulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

A. Poveda Rosenthal compacta that are premetric of finite de-gree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

A. Prochazka On low-distortion embeddings of metric spacesinto reflexive spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

D. J. Rodrıguez Translation-invariant subspaces for weighted L2

on R+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3

J. Seoane Isomorphic copies of `1 for m-homogeneous non-analytic Bohnenblust-Hille polynomials . . . . . . . . . . 25

F. Strobin Spaceability of sets in Lp × Lq and C0 × C0 . . . . . 25

P. Tradacete Weakly compactly generated Banach lattices . . . . . 26

H. Yurt Approximation of Several Variables Functions inLorentz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

PLENARY TALKS

7

Algebras of symmetric analytic functions (bis)Richard M. Aron (Kent State University — USA)

Let X be a complex Banach space with basis {en} and open unit ball B. Wealso suppose that the basis is symmetric, that ‖

∑n anen‖ = ‖

∑n aσnen‖

for every permutation σ on N and every∑

n aσnen ∈ X. In a number ofpapers, we have-with colleagues-studied C−valued holomorphic functionsf on X that are symmetric; that is, f(

∑n anen) = f(

∑n aσnen) for all

convergent∑

n anen and all permutation σ.

In this talk, we will describe new, related work in two directions:(i) Holomorphic functions f : B → B (or f : X → X) having the propertythat for any element g in a group G of holomorphic mappings B → B (orX → X), f = f ◦ g;(ii) Symmetric holomorphic functions on D2 or B2 in C2. This second partcan be regarded as a special case of (i), in which G is the two element groupconsisting of the identity and the map (z1, z2) ; (z2, z1). This part is alsorelated to work of Agler and Young on the symmetrized bidisc.

Polynomial ideals from a nonlinear viewpointGeraldo Botelho (Universidade Federal de Uberlandia – Brazil)

Classes of homogeneous polynomials between Banach spaces have been stud-ied in the last three decades from the perspective of the so-called ideal prop-erty: if a polynomial P belongs to a class Q, then the composition u ◦P ◦ vof P with linear operators u and v belongs to Q as well. In an attempt toexplore the nonlinearity of the subject in a more consistent way, and takinginto account recent results in the field, we propose the study of classes ofhomogeneous polynomials that are stable under the composition with ho-mogeneous polynomials. Some important classes justify the study of theintermediate concept of classes of polynomials Q such that if P belongs toQ, u is a linear operator and Q is a homogeneous polynomial, then u◦P ◦Qbelongs to Q.

This talk is based on a joint work with Ewerton R. Torres.

8

Lineability and additivity cardinals for real-valuedfunctions: old results and new developmentsKrzysztof Chris Ciesielski(Department of Mathematics, West Virginia University, USA,

and

Department of Radiology, University of Pennsylvania, USA)

In the talk we will discuss several cardinal coefficients related to the linearstructure of the classes of functions from X to which, in their entirety, donot form linear spaces. In particular, we will examine: the additivity numberA(M), defined as the minimum cardinality of a family F ⊂X for which thereis no f ∈X with f + F ⊂ M ; the lineability number L(M), defined as thesmallest cardinal number κ for which M ∪ {0} contains no vector space ofdimension κ; as well as the maximal lineability, mL(M) and homogeneouslineability, HL(M) numbers. We will describe the relations between thesecoefficients and give their values for some well studied classes of functions.

Self-contracted curves: from Euclidean spaces toRiemannian manifoldsAris Daniilidis (Universidad de Chile — Chile)

Self-contracted curves have been initially introduced for the study of (dis-crete or continuous) steepest descent systems of quasiconvex functions inHilbert spaces. However, their definition is simple and is naturally formu-lated in mere metric spaces, without any prior regularity assumption (con-tinuity or absolute continuity) of the curve. This has motivated a study ofintrinsic properties of such curves, mainly related to rectifiability.

In this talk, which is based on joint works with G. David (Orsay),R. Deville (Bordeaux), E. Durand (UNED), A. Lemenant (Paris7) and L. Rifford (Nice) we shall describe the state-of-the-art onthis topic.

9

Embedding metric spaces into c0

Robert Deville (Universite de Bordeaux — France)

Let (M,d) be a separable metric space and λ > 1. We say that M stronglyλ-embeds into c0 (endowed with its usual supremum norm) if there existsf = (fn) : M → c0 Lipschitz continuous, such that whenever x, y ∈ M andx 6= y, then d(x, y) < ‖f(x) − f(y)‖ and for each n, the Lipschitz constantof fn is less than λ. We characterize separable metric spaces that stronglyλ-embed into c0 by an internal property called π(λ). All separable metricspaces strongly 2-embed into c0.

This is a joint work with Florent Baudier and simplifies formerwork obtained by Nigel Kalton and Gilles Lancien.

On the structure of Pw(nE,F ) as a subspace ofP(nE,F )Veronica Dimant (Universidad de San Andres — Argentina)

The space P(nE,F ) of continuous n-homogeneous polynomials between Ba-nach spaces E and F has a distinguished subspace: Pw(nE,F ) that consistsof those polynomials which are weakly continuous on bounded sets. In thistalk we discuss the problem of when this special subspace is an M -ideal inits ambient space. We present some consequences of this fact and provide anumber of examples. We also consider some weaker structures like to be anHB-subspace or an M(1, C)-ideal.

Part of this talk comes from joint work with Silvia Lassalle andAngelines Prieto.

10

Preservation of Poincare inequalities undersphericalization and flattening in the metric settingEstibalitz Durand Cartagena (UNED— Spain)

The process of obtaining the Riemann sphere from the complex plane, andvice versa, was generalized in the metric setting by using sphericalizationand flattening. These conformal transformations are dual to each other,and the performance of sphericalization followed by flattening, or vice versa,results in a metric space that is bi-Lipschitz equivalent to the original space.A very natural problem is therefore to study which geometric propertiesare preserved under these transformations.

Metric spaces endowed with a doubling measure and supporting a Poincareinequality are nowadays considered a standard class of spaces when devel-oping a first order differential analysis in a metric measure space setting.In this talk we will focus on the preservation of Poincare inequalities undersphericalization and flattening.

Part of the talk is based on a joint work with Xining Li.

Continuous selections of Lipschitz extensions inBanach and metric spacesRafael Espınola (Universidad de Sevilla — Spain)

In this talk we review some results on extension of Lipschitz mappings onBanach and metric spaces in order to describe new results on parameterdependence of these extensions from the point of view of continuity. That is,we study when Lipschitz extensions of Lipschitz mappings can be obtainedand chosen in such a way that extensions depend on a continuously way onthe extended mappings. We deal with these problems in metric spaces withappropriate curvature bounds which will also be described during the talk.Our approach has been motivated by recent results by Kopecka and Reichon Hilbert spaces.

Joint work with Adriana Nicolae.

11

A Radon transform on infinite dimensions: the fate ofTomographyAlberto Ibort (Universidad Carlos III de Madrid — Spain)

The Radon transform provides the mathematical background for standardtomography. An extension of it to complex separable infinite-dimensionalHilbert spaces inspired on harmonic analysis on groups will be discussed.An application to the tomography of quantum systems will be described.

Strongly weakly compactly generated Banach spacesand their relativesSebastian Lajara (Universidad de Castilla-La Mancha — Spain)

In this talk, we survey the general properties of strongly weakly compactlygenerated Banach spaces, and present several results about the relatedclasses of strongly Asplund generated and strongly conditionally weaklycompactly generated spaces. We show that every strongly Asplund gen-erated space is strongly conditionally weakly compactly generated, and thata Banach space is strongly weakly compactly generated if, and only if, it isweakly sequentially complete and strongly conditionally weakly compactlygenerated. We also prove that the notions of strongly Asplund generated andstrongly weakly conditionally compactly generated space coincide for Ba-nach lattices. Some applications in the setting of Lebesgue-Bochner spacesare also given.

The talk is based on a joint work with J. Rodriguez (Universidadde Murcia).

Regularity of 1D almost minimal sets in Banach spacesAntoine Lemenant (Universite Paris Diderot — France)

I will talk about a joint work with Thierry De Pauw (Paris 7, IMJ) andVincent Millot (Paris 7, LJLL), which contains some local regularity resultsfor 1D almost minimal sets in Banach spaces. A 1D-almost minimal set is aclosed and connect set that locally minimizes the Hausdorff measure H1, upto an excess controlled by some gauge function ξ(r) in a ball of radius r. Itis well known that such sets are C1 regular almost everywhere in euclideanspaces, provided that the gauge is Dini integrable. In this talk I will presenta condition on the ambient norm for which this result still holds true in aBanach space. Some questions about the optimality of this condition willbe also discussed.

12

The algebra of the ballManuel Maestre (Universitat de Valencia — Spain)

On this talk we will survey on the infinite dimensional counterpart of thealgebra of the disk A(D) of all complex functions continuous on the closedunit disk D and holomorphic in its interior. It is a kind of crossover ofcomplex analysis and Banach theory. On the one hand we will discuss thesize and structure of the maximal ideal space of the algebra of the ball ofsome classical Banach spaces as c0, `1 and `2. On the other, we will surveyon properties of interest in Banach spaces as Daugavet property, numericalradius and the Schur property.

Spaces of compact operatorsDaniel Puglisi (Universita degli Studi di Catania — Italy)

It has been a long standing question to see if the only bounded linearprojection from the space L(X;Y ) of continuous linear operators ontothe space K(X;Y ) of compact linear operators is the trivial one (i.e.,L(X;Y ) = K(X;Y )). From one hand, it is already well known that ifc0 embeds in K(X;Y ), then K(X;Y ) is uncomplemented in L(X;Y ). Onthe other hand, in this talk we construct a Banach space X such that K(X)is uncomplemented in L(X) even if K(X) does not contains c0. In the sec-ond part of this talk, we investigate the family of separable Banach spaces(X;Y ) such that K(X;Y ) is complemented in L(X;Y ) from a DescriptiveSet Theory point of view. Moreover, we show how the space of compactoperator can be involved even to construct space of continuous functions.

Part of this talk is a jointly work with P. Motakis and D. Zisi-mopoulou.

SHORT TALKS

15

Morse and Sard meet Taylor and StepanovDaniel Azagra (Universidad Complutense de Madrid — Spain)

Let n,m, k be positive integers with k = n−m+1. We establish an abstractMorse-Sard-type theorem which allows us to deduce, on the one hand, a pre-

vious result of De Pascale’s for Sobolev W k,ploc (Rn,Rm) functions with p > n

and, on the other hand, also the following new result: if f ∈ Ck−1(Rn,Rm)has a Taylor expansion of order k at almost every point x ∈ Rn and satisfies|f(x + h) − f(x) − Df(x)(h) − ... − 1

(k−1)!Dk−1f(x)(hk−1)| = O(|h|k) for

every x ∈ Rn, then the set of critical values of f is Lebesgue-null in Rm.In particular this allows us to establish the Morse-Sard property for severalclasses of functions with badly behaved derivatives of order k − 1 (for in-stance, functions whose derivatives of order k − 1 are pointwise Lipschitz,AKA Stepanov functions) to which none of the previous generalizations ofthe Morse-Sard theorem can be applied.

This is joint work with Juan Ferrera and Javier Gomez-Gil.

Everywhere surjective and Sierpinski-ZygmundfunctionsMarek Bienias (Institute of Mathematics Lodz University ofTechnology, Institute of Mathematics, University of Gdansk —Poland)

During the talk, I will introduce the notion of an everywhere κ surjectivefunction and recall the notion of Sierpinski-Zygmund function: in the senseof continuous and Borel mappings. The main part of the talk will be theproofs of the following facts:

• the set of everywhere but not strongly everywhere surjective complexfunctions is strongly c-algebrable and its 2c-algebrability is consistentwith ZFC• under CH the set of everywhere surjective complex functions which

are Sierpinski-Zygmund in the sense of continuous but not Borelfunctions is strongly c-algebrable.

joint work with A. Bartoszewicz, S. G lab and T. Natkaniec.

16

Optimal Control Problem For Processes Given ByMulti-Parameter Linear Stochastic Dynamic SystemMuhammet Candan (Canakkale Onsekiz Mart University — Turkey)

In this work, optimal control problem for processes represented by multi-parameter linear stochastic dynamic system is investigated (1) and (2). Also,for existence of optimal control and corresponding optimal trajectory, the-orem of necessity and sufficiency condition for the considered problem isproven.

References(1) I. V. Gaishun, Completely Solvable Multidimensional Differential Equations,

Nauka, (1983), 231.(2) R. Gabasov, F. M. Kirillova, N. S. Paulianok, Optimal Control of Linear Systems

on Quadratic Performance Index, Appl. and Comput. Math.7 (2008) 4-20.(3) Y. Hac, K. Ozen, Terminal Control Problem for Represented by Nonlinear Multi-

parameter binary Dynamic System, Control and Cybernetics. 38 (2009)625-633.(4) V. G. Boltyanskii, Optimal control of Discrete Systems, JohnWilley, (1978), 363.(5) S.V. Yablonsky, Introduction to Discrete Mathematics, Mir Publisher, Moscow,

9.

Joint work with Yakup Haci.

On Lipschitz-free spaces of ultrametric spacesAude Dalet (Laboratoire de Mathmatiques de Besancon— France)

Let M be a pointed metric space and Lip0(M) the space of Lipschitz func-tions vanishing at 0. Endowed with the Lipschitz norm this space is aBanach space. Denote F(M) the closed subspace of Lip0(M)∗ spanned bythe evaluation points and call it the Lipschitz-free space over M .

We will first study Lipschitz-free spaces over compact ultrametric spaces andprove that they are dual spaces isomorphic to `1, with a predual isomorphicto c0. However, we will then prove that Lipschitz-free spaces over ultrametricspaces are never isometric to `1.

17

Porosity of contractions in spaces of non-expansivemappingsMichael Dymond (The Technion - Israel Institute of Technology.Universitat Innsbruck — Austria)

We consider a large class of geodesic metric spaces, including Banach spaces,hyperbolic spaces and geodesic CAT(κ)-spaces, and investigate the space ofnonexpansive mappings on either a convex or a star-shaped subset in thesesettings. We prove that the strict contractions form a negligible subset ofthis space in the sense that they form a σ-porous subset. For separablemetric spaces we show that a generic nonexpansive mapping has Lipschitzconstant one at typical points of its domain. These results contain thecase of nonexpansive self-mappings and the case of nonexpansive set-valuedmappings as particular cases.

Joint work with Christian Bargetz and Simeon Reich (The Tech-nion - Israel Institute of Technology Universitat Innsbruck).

18

Asymptotic uniform smoothness in spaces of compactoperatorsLuis C. Garcıa-Lirola (Universidad de Murcia — Spain)

The modulus of asymptotic uniform smoothness of a Banach space X isgiven by

ρX(t, x) = supx∈SX

infdim(X/Y )<∞

supy∈SY

||x+ ty|| − 1 .

The space X is said to be is said to be asymptotically uniformly smooth (AUSfor short) if limt→0 t

−1ρX(t) = 0. It is well-known that a Banach space isAUS if and only if its dual space is weak* uniformly Kadec-Klee. Lennardproved in [3] that the space of compact operators K(`2, `2) is AUS. As aconsequence, the space nuclear operators N (`2, `2) has the weak* fixed pointproperty. This result was extended by Besbes [1] to K(`p, `q) with p−1 +q−1

= 1. Moreover, it is shown in [2] that the same is true for 1 < p, q < 1.Among other results, we show that K(X,Y ) is AUS for more general spacesX and Y , including Orlicz sequence spaces.

References

[1] M. Besbes: Points fixes dans les espaces des oprateurs nucleaires, Bull. Austral.Math. Soc. 46 (1992), no. 2, 287-294.

[2] S.J. Dilworth, D. Kutzarova, N. Lovasoa Randrianarivony, J.P. Revalski,N.V. Zhivkov: Compactly uniformly convex spaces and property (β) of Rolewicz,J. Math. Anal. Appl. 402 (2013), no. 1, 297-307.

[3] C. Lennard: C1 is uniformly Kadec-Klee, Proc. Amer. Math. Soc. 109 (1990), no.1, 71-77.

This is part of an ongoing work with Matıas Raja. This research ispartially supported by the grants MINECO/FEDER MTM2014-57838-C2-1-P and Fundacion Seneca CARM 19368/PI/14.

Asymptotic smoothness, convex envelopes andpolynomial normsRaquel Gonzalo (Universidad Politecnica de Madrid — Spain)

In this talk we introduce a suitable notion of asymptotic smoothness oninfinite dimensional Banach spaces and we prove that under some structuralrestrictions on the space, the convex envelope of an asymptotically smoothfunction is asymptotically smooth. Furthermore, we compute the best orderof the modulus of convexity of a polynomial norm.

Joint work with J. A. Jaramillo (Universidad Complutense deMadrid) and Diego Yanez (Universidad de Extremadura).

19

Completeness of Mackey Topologies on BanachSpacesAntonio J. Guirao (Universitat Politecnica de Valencia — Spain)

Some variations of an example on Mackey topologies in Banach spaces pro-vided by J. Bonet and B. Cascales are given by using a simple result onMazur spaces. Moreover, we fully characterize the situation (completenessor the lack of it) in the case of Banach spaces with a w∗-angelic dual unit ballby using the concept of norming subspace. A related question concerningthe space c0 is answered. Some other applications are provided.

Joint work with Vicente Montesinos and Vaclav Zizler.

On the set of limit points of conditionally convergentseriesJacek Marchwicki (Institute of Mathematics, Lodz University ofTechnology — Poland)

Let∑∞

n=1 xn be a conditionally convergent series in a Banach space and let τbe a permutation of natural numbers. We study the set LIM(

∑∞n=1 xτ(n))

of all limit points of a sequence (∑p

n=1 xτ(n))∞p=1 of partial sums of a re-

arranged series∑∞

n=1 xτ(n). We give full characterization of limit sets infinite dimensional spaces. Namely, a limit set in Rm is either compact andconnected or it is closed and all its connected components are unbounded.On the other hand each set of one of these types is a limit set of somerearranged conditionally convergent series. Moreover, this characterizationdoes not hold in infinite dimensional spaces. We show that if

∑∞n=1 xn has

the Rearrangement Property and A is a closed subset of the closure of the∑∞n=1 xn sum range and it is ε-chainable for every ε > 0, then there is a

permutation τ such that A = LIM(∑∞

n=1 xτ(n)). As a byproduct of thisobservation we obtain that series having the Rearrangement Property haveclosed sum ranges.

Joint work with Szymon G lab (Institute of Mathematics, LodzUniversity of Technology).

20

More on Lipschitz-type functionsAna Soledad Merono (Universidad Complutense de Madrid — Spain)

Recall that a real function f on a metric space (X, d) is called CauchyLipschitz if it is Lipschitz when restricted to the totally bounded subsetsof X. Moreover, f is uniformly locally Lipschitz if there exists some δ > 0such that f is Lipschitz when restricted to the open balls Bδ(x) of radiusδ for every x ∈ X. In spite of having that the locally Lipschitz are thosefunctions that are Lipschitz of the compact subsets, we have that in generalCauchy Lipschitz functions and uniformly locally Lipschitz functions aredifferent families of functions [1], [2]. In this talk we introduce a couple ofnew families of Lipschitz-type functions, additionally to the previous ones,by beans of the family of the Bourbaki-bounded sets and the consecutiveδ-enlargements Bm

δ (x) of the balls of radius δ > 0. In particular, we provethat the equality between all these families characterize different types ofstronger metric completeness [4] and those spaces in which the family or realuniformly continuous functions is a ring [3].

References

[1] G. Beer, M. I. Garrido, Locally Lipschitz functions, cofinal completeness, and UCsapces, Jour, Math. Anal. Appl. 428 (2015) 804-816.

[2] G. Beer, M. I. Garrido, On the uniform continuity of Cauchy continuous functions,submitted.

[3] J. Cabello, When is U(X) a ring? submitted.[4] M. I. Garrido, A. S. Merono, New types of completeness in metric spaces, Ann. Acad.Sci. Fenn. Math. 39 (2014) 733-758.

Joint work with M. I. Garrido (Universidad Complutense deMadrid).

Interpolating sequences for Bloch spaces on the unitball of a Hilbert spaceAlejandro Miralles (Universitat Jaume I de Castello — Spain)

We extend the study of interpolating sequences for spaces of analytic func-tions to the case when we deal with the Bloch space of an infinite dimensionalHilbert space.

21

Topological and algebraic properties of spaces ofLorch analytic mappingsLuiza A. Moraes (Universidade Federal do Rio de Janeiro (UFRJ) —Brasil)

If E is a commutative complex Banach algebra with unit and U is an open(non empty) connected subset of E, a mapping f : U ⊂ E → E is Lorchanalytic in U if given any a ∈ U there exists ρ > 0 and there exist uniqueelements an ∈ E, such that Bρ(a) ⊂ U and f(z) =

∑∞n=0 an(z − a)n, for

all z in ‖z − a‖ < ρ. The theory of Lorch analytic mappings goes back tothe 1940’s (cf. [4]) and is a very natural extension of the classical conceptof analytic function to infinite dimensional algebras. Let f ∈ HL(U,E)(the space of all Lorch analytic mappings in an open subset U of E) andφ ∈ M(E) (the set of all complex homomorphisms not identically 0 on E).If there exists a (necessarily unique) complex analytic function g : φ(U)→ Cso that g ◦ φ = φ ◦ f on U, we say that g is the quotient function of f withrespect to φ and write g = fφ. Glickfeld defined the quotient function in[1]. In [3] he discussed the existence of fφ and used this idea to prove twodifferent versions of the inverse function theorem (under different specialhypothesis). By using the idea of quotient function, he was also able toobtain a generalization of the Mittag-Leffler’s Theorem in [2].In this talk we will present algebraic and topological results concerningHL(U,E) endowed with convenient topologies. Many of these results havebeen obtained by using the idea of quotient function f ∈ HL(U,E) withrespect to φ ∈M(E).

REFERENCES

[1] B. W. Glickfeld, The Riemann sphere of a commutative Banach algebra,Trans. Amer. Math. Soc. 134 (1968) 1-28.[2] B. W. Glickfeld, Meromorphic functions of elements of a commutativeBanach algebra, Trans. Amer. Math. Soc. 151 (1) (1970) 293-307.[3] B. W. Glickfeld, On the inverse function theorem in commutative Banachalgebras, Illinois J. Math. 15 (1971) 212-221.[4] E. R. Lorch, The theory of analytic functions in normed abelian vectorrings, Trans. Amer. Math. Soc. 54 (1943), 414–425.

Joint work with Guilherme V. S. Mauro (IUniversidade Federalda Integracao Latino-Americana (UNILA)) and Alex F. Pereira(Universidade Federal Fluminense (UFF)).

22

Successive Approximation Method in Matrix GamesAykut Or (Canakkale Onsekiz Mart University,— Turkey)

Let X Banach space and A be a linear operator from X to X . Succes-sive Approximation Method (SAM) is a method used commonly solutionof equation specified as x = Ax. and SAM is used in finding solutions oflinear equation systems. Besides, in matrix games payoff of players can bespecified in the form of linear equation systems. After we solve n× n linearequation systems with the help of SAM, we obtain solution of the game andmixed strategies of players.

References(1) Kreyszig E., 1978. Introductory Functional Analysis with Applications. John

Wiley and Sons 209-299.(2) Barron, E. N., 2008. Game Theory an Introduction. John Wiley and Sons, Inc.,

New Jersey. 1-108.(3) Owen G., 1995. Game Theory. Academic Press, Inc. 1-85.

Joint work with Muhammet Candan and Yakup Haci (CanakkaleOnsekiz Mart University, Canakkale, Turkey).

A Dissipative System Driven By A Minimal TurbulenceVictor Payne (University Of Ibadan — Nigeria)

A Non-Linear Partial Differential Equation Describing A Dissipative SystemDriven By A Minimal Turbulence Is Considered Using Harmiltonia SystemIn Which Question Of Long Time Behaviour For Energy Transfer And Dis-tribution Is Discussed.

Joint work with G.S Lawal (University Of Ibadan).

23

Rosenthal compacta that are premetric of finite degreeAlejandro Poveda (Universidad de Murcia — Spain)

A compact space K is said to be Rosenthal if it is homeomorphic to apointwise compact space in B1(X) where X is a polish space. Rosenthalcompacta have become an important research topic on Functional Analysisafter the well-known Odell-Rosenthal’s characterization of separable Banachspaces without copies of `1.

Despite its analytical origins, Rosenthal compacta have been extensivelystudied from other areas like general topology and its study involves tech-niques from Logic, Ramsey Theory and Descriptive Set Theory. In thisregard we can highlight a celebrated result on the structure of the class ofseparable Rosenthal compacta due to our third author. Namely, Todorce-vic proved that a separable Rosenthal compactum either contains a discretesubspace of size continuum or it is a pre-metric compactum of degree at mosttwo. Moreover, the author proves that such kind of compacta contain eithercopies of the Split Interval S(I) or copies of the Alexadroff Duplicate of theCantor space D(2N).

In this work we generalize the idea of pre-metric compactum of degree atmost two considering the notion of Rosenthal compactum of finite degree andinvestigate the structure of this new class of Rosenthal compacta. Specifi-cally, we prove that every separable Rosenthal compactum of degree n butnot of degree n − 1 contain either copies of the n−Split Interval Sn(I) orcopies of the Alexadroff n− plicate Dn(2N)

Joint work with Antonio Aviles (Universidad de Murcia) andStevo Todorcevic (University of Toronto-CNRS).

24

On low-distortion embeddings of metric spaces intoreflexive spacesAntonın Prochazka (Universite de Franche-Comte — France)

We say that a metric space M Lipschitz-embeds with distortion D ≥ 1 intoa Banach space X, and we use the symbol M ↪→

DX, if there is f : M → X

such that d(x, y) ≤ ‖f(x)− f(y)‖ ≤ Dd(x, y). We will discuss the followingand similar theorems.

Theorem. 1. There exists a countable metric graph M such that for everyBanach space X with an unconditional basis the following is equivalent.

(1) X is not reflexive,(2) there is an equivalent norm | · | on X such that M ↪→

1(X, | · |),

(3) there is an equivalent norm | · | on X and D < 2 s.t M ↪→D

(X, | · |).

Translation-invariant subspaces for weighted L2 on R+

Daniel J. Rodrıguez (Universidad Complutense de Madrid — Spain)

In the setting of weighted L2-spaces, a striking result due to Domar statesthat the lattice of closed invariant subspaces for {Sτ}τ≥0 in L2(R+, w(t)dt)coincides with the lattice of “standard invariant subspaces”

L2([a,∞), w(t)dt) = {f ∈ L2(R+, w(t)dt) : f(t) = 0 a.e 0 ≤ t ≤ a}, (a ≥ 0),

whenever w satisfies:

(1) w positive continuous decreasing in R+.

(2) logw is concave in [c,∞), for some c ≥ 0.

(3) limt→∞

− logw(t)

t=∞ and lim

t→∞

log | logw(t)| − log t√log t

=∞.

In this talk we present an extension of Domar’s Theorem to a wider classof weight functions w not fulfilling condition (2), which is replaced by ageometric condition on {w(tn)}n≥1 for some strictly increasing sequence{tn}n≥1 ⊂ R+ with the uniformly bounded condition supn(tn+1 − tn) <∞.This extension addresses, in some sense, a question posed by Domar.In addition, we provide an example of a weight function w for which thelattice of closed invariant subspaces for {Sτ}τ≥0 in L2(R+, w(t)dt) is non-standard such that w satisfies (3) and the geometric condition is fulfilledonly for sequences {tn}≥1 ⊂ R+ with supn(tn+1 − tn) =∞

Joint work with Eva A. Gallardo-Gutıerrez (Universidad Com-plutense de Madrid, Spain) and Jonathan R. Partington (Univer-sity of Leeds, UK).

25

Isomorphic copies of `1 for m-homogeneousnon-analytic Bohnenblust-Hille polynomialsJuan B. Seoane Sepulveda (Universidad Complutense de Madrid -ICMAT — Spain)

We employ a classical result by Toeplitz (1913) and the seminal work byBohnenblust and Hille on Dirichlet series (1931) to show that the set ofcontinuous m-homogeneous non-analytic polynomials on c0 contains an iso-morphic copy of `1. Moreover, we can have this copy of `1 in such a waythat every non-zero element of it fails to be analytic at precisely the samepoint.

The material presented in the lecture is part of a joint work withJ. Alberto Conejero and Pablo Sevilla-Peris (IUMPA - UniversitatPolitecnica de Valencia).

Spaceability of sets in Lp × Lq and C0 × C0

Filip Strobin (Institute of Mathematics of the Lodz University ofTechnology — Poland)

A subset E of an infinitely dimensional linearly–topological space X is calledspaceable if there is an infinitely dimensional closed subspace Y of X withY ⊂ E ∪{0}. During the talk I will present the spaceability of the followingsets:

1. the set of those (f, g) ∈ Lp × Lq for which fg /∈ Lr provided thatone of the following conditions holds:(a) 0 < 1

p + 1q <

1r and sup{µ(A) : µ(A) <∞} =∞

(b) 1p + 1

q >1r and inf{µ(A) : µ(A) > 0} = 0;

2. the set of those (f, g) ∈ C0×C0 for which fg is not integrable, whereC0 is the space of continuous mappings which vanish at infinity;

3. the set of those (f, g) ∈ Lp(G) × Lq(G) for which the convolutionf ?g is not well-defined or is infinite, provided G is a locally compactnon-compact topological group and p, q > 1 with 1/p+ 1/q < 1.

The results are related to our previous ones [GS1], [GS2], [GS3] in which westudied these sets from the Baire category and σ-porosity points of view.

REFERENCES

[GS1] G lab, S.; Strobin, F. Dichotomies for Lp spaces. J. Math. Anal.Appl., 368 (2010) 382–390.[GS2] G lab, S.; Strobin, F. Porosity and the Lp-conjecture. Arch. Math. 95(2010), 583–592.[GS3] G lab, S.; Strobin, F. Dichotomies for C0(X) and Cb(X) spaces,Czechoslovak Math. J., 63 (138) (2013), no. 1, 91–105.

Joint work with Szymon G lab (Institute of Mathematics of the Lodz University of Technology, Poland).

26

Weakly compactly generated Banach latticesPedro Tradacete (Universidad Carlos III de Madrid — Spain)

We study the different ways in which a weakly compact set can generate aBanach lattice. Among other things, it is shown that in an order continuousBanach lattice X, the existence of a weakly compact set K ⊂ X such thatX coincides with the band generated by K, implies that X is WCG

Joint work with A. Aviles (Universidad de Murcia), A.J. Guirao(Universitat Politecnica de Valencia), S. Lajara (Universidad deCastilla-La Mancha) and J. Rodrıguez (Universidad de Murcia).

Approximation of Several Variables Functions InLorentz SpacesHasan Yurt (Canakkale Onsekiz Mart University — Turkey)

The multivariate Lorentz spaces, which was generalization of Lebesguespaces, were introduced by Blozinski in 1981. In his paper, he investi-gated some problems in these spaces. In this study, we will give Jacksontype direct and Berstein-Stechkin-type converse theorems for trigonometricpolynomials in Lorentz spaces, in the case of several variables. In particular,we will obtain a constructive characterization of the generalized Lipschitzclasses in these spaces.