Madrid, 7–10 February 2012 • Facultad de Ciencias Matematicas • Universidad Complutense de Madrid • SPAIN

FUNCTION THEORY ONINFINITE DIMENSIONAL SPACESXIIOrganizing committee: D. Azagra, E. Durand, J. L. Gamez, G. A. Munoz, L. F. Sanchez, J. B. Seoane

BOOK OF

ABSTRACTS

FT DS2012

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

Plenary TalksR. Aron Fiber structure for H∞-functions . . . . . . . . . . . . . . . 7

A. Daniilidis Integration of multivalued operators in the light ofconvex and nonsmooth analysis . . . . . . . . . . . . . . . . . 7

R. Deville Construction of an operator with wild dynamics . 7

E. Durand Rectifiable curves in Sierpinski carpets . . . . . . . . . . 8

D. Garcıa The Bishop-Phelps-Bollobas Theorem . . . . . . . . . . . 9

G. Godefroy Free spaces over compact metric spaces . . . . . . . . . . 9

L. A. Harris Derivatives of bivariate polynomials, Markov’stheorem and Geronimus nodes . . . . . . . . . . . . . . . . . . 10

Y. Kinnunen Mapping properties of the discrete maximal oper-ator in metric measure spaces . . . . . . . . . . . . . . . . . . . 10

G. Lancien Non linear quotients and asymptotic uniformstructure of Banach spaces . . . . . . . . . . . . . . . . . . . . . . 11

L. Lindstrom Weighted composition operators on Bloch typespaces: new estimates of the essential norm . . . . . 11

M. Maestre Monomial expansions of Hp-functions in infinitelymany variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

D. Pellegrino New advances and directions in the study of abso-lutely summing (linear and nonlinear) operators 12

A. Seeger Functional analysis and geometry of convex conesin Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

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Short TalksA. Aviles A continuous image of a Radon-Nikodym compact

space which is not Radon-Nikodym . . . . . . . . . . . . . . 17

G. Botelho Nicodemi sequences of operators between spaces ofmultilinear mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

G. J. Buskes Polynomials and positive tensor products . . . . . . . . 17

J. A. Conejero Devaney chaos and distributional chaos in the so-lutions of certain partial differential equations . . . 18

M. Fabian Uniform Eberlein compacta and coincidence ofPettis and McShane integrability . . . . . . . . . . . . . . . . 18

V. Favaro Maximal spaceability in topological vector spaces . 19

D. E. Galicer Geometry of integral polynomials, M -ideals andunique norm preserving extensions . . . . . . . . . . . . . . 19

Sz. G lab Algebrability and strong algebrability . . . . . . . . . . . . 20

J. Gong Metric currents in the plane . . . . . . . . . . . . . . . . . . . . 20

E. Jorda Weighted Banach spaces of harmonic functions . . 21

P. Kaufmann Spaceability of sets of nowhere Lq functions . . . . . 22

M. Lacruz Hardy-Littlewood inequalities for norms of positiveoperators on sequence spaces . . . . . . . . . . . . . . . . . . . . 22

S. Lajara Smooth renormings of the Lebesgue-Bochner func-tion space L1(µ,X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

J. Lopez-Salazar Lineability of the set of holomorphic mappingswith dense range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

F. Martınez The specification property in the dynamics of lin-ear operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

M. Mazzitelli On the polynomial Lindenstrauss theorem . . . . . . . 25

A. Miralles Hypercyclic composition operators on H0v -spaces . 26

L. A. de Moraes Topological and algebraic properties of spaces ofLorch analytic mappings . . . . . . . . . . . . . . . . . . . . . . . . 27

S. Muro Algebras of bounded type functions associated to aholomorphy type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

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O. Nygaard Diameter 2 properties in Banach spaces . . . . . . . . . 28

D. Pinasco Lower bounds for norms of products of polynomi-als on `p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

J. Rodrıguez Measurability in C(2κ) and Kunen cardinals . . . . . 30

P. Rueda Projective tensor products of Lp-spaces . . . . . . . . . . 30

L. Sanchez Reproducing kernel Banach spaces and learningmachines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

J. Santos A general version of the Pietsch Domination The-orem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Y. Sarantopoulos Polarization constants and the plank problem . . . . 31

E. Saukko Differences of composition operators betweenBergman spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

P. Tradacete Spectral properties of disjointly strictly singularoperators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

P. Turco On p-compact polynomials and holomorphic map-pings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

D. Yost Contractive projections on Banach spaces . . . . . . . 33

PLENARY TALKS

7

Fiber structure for H∞-functionsRichard M. Aron (Kent State University — USA)

This will largely be an expository talk in which work of Farmer [1] and Cole,Gamelin, and Johnson [2] will be reviewed, along with more recent work bythe author with Carando, Gamelin, Lassalle, and Maestre [3]. In addition,mention will be made of new work concerning fibers of H∞(Bc0) over theboundary of B`∞ .

Let E be a complex Banach space with open unit ball BE . The algebrathat we’ll primarily deal with is H∞(BE) ≡ {f : BE → C | f is holomorphicand bounded on BE}, which is a Banach algebra when endowed with thesup norm. Let M(BE) ≡ {ϕ : H∞(BE) → C | ϕ is a homomorphism }.Considering each homomorphism as acting on elements of E∗ ⊂ H∞(BE),we can regard each such ϕ as lying over a point z∗∗ ∈ BE∗∗ . In a naturalsense, every function f ∈ H∞(BE) extends to f : M(BE) → C, and soone can study analytic structure of M(BE). In some cases, there is evena rich structure in the ‘fiber’ consisting of all homomorphisms lying over apoint z∗∗.

Our aim is to review this interesting area and to discuss some possiblenew local results.

References

[1] J. D. Farmer, Fibers over the sphere of a uniformly convex Banach space, Mich.Math. J. 45 (1998), no.2, 211–226.

[2] B. J. Cole, T. W. Gamelin, and W. B. Johnson, Analytic discs in fibers over theunit ball of a Banach space, Mich. Math. J. 39 (1992), no. 3, 551–569.

[3] R. M. Aron, D. Carando, T. W. Gamelin, S. Lassalle, and M. Maestre,Cluster values of analytic functions on a Banach space, Math. Annalen, to appear.

Integration of multivalued operators in the light ofconvex and nonsmooth analysisAris Daniliidis (Universitat Autonoma de Barcelona — Spain)

In this talk we discuss the following problem: given a Banach space X anda multivalued operator T : X ⇒ X∗, find a nonsmooth primitive functionf : X → R ∪ {+∞} such that T = ∂f, where ∂ is the Clarke subdifferen-tial. A cornerstone result of Rockafellar in convex analysis gives a completeanswer in the case T is maximal cyclically monotone, constructing explic-itly its (unique up to a constant) lower semicontinuous convex primitive.Enhancing on this construction we obtain a generalization corresponging tothe class of approximately convex functions (lower-C1, in case X = Rn).

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Construction of an operator with wild dynamicsRobert Deville (Universite de Bordeaux — France)

The theory of hypercyclic operators (i.e., operators such that there existsa point with dense orbit) show that the linear dynamic can be quite com-plicated. We present a new construction of operators due to J. M. Auge.More precisely, for every separable Banach space X, there exists a boundedlinear operator T on X such that for every x ∈ X, either the sequence(‖Tnx‖) tends to infinity or lim inf‖Tnx− x‖ = 0 (i.e., the point x is recur-rent for T ), and each of these cases occur on a set of non empty interior.Moreover, Id−T is a compact operator.

Rectifiable curves in Sierpinski carpetsEstibalitz Durand Cartagena (Universidad Nacional de Educacion aDistancia — Spain)

In the last years, there has been an intensive research on the setting of metricmeasure spaces, where a first order differential calculus has been developed.Standard assumptions in analysis on metric spaces include that the measureis doubling and the space supports a p-Poincare inequality. In some sense,these conditions guarantee that any pair of points can be connected by afamily of curves that are not too long and that the curves can be nicelydistributed.

In this talk we study a particular case of doubling metric measure space.We focus our attention in a classical fractal: the Sierpinski carpet endowedwith its associated Hausdorff measure.

In the first part of the talk, we will review some of the latest results whichhave contributed to understand the geometrical structure of metric measurespaces supporting a p-Poincare inequality and explain why the families ofcurves that live in the Sierpinski carpet are not enough for our purposes;that is, in terms of Poincare inequalities.

In the second part, we will characterize the slopes of nontrivial line seg-ments contained in self-similar Sierpinski carpets. The set of slopes is relatedto Farey sequences and the dynamics of punctured square toral billiards. Asa consequence, we deduce conclusions about the collection of everywheredifferentiable curves contained in such carpets.

Joint work with J. A. Jaramillo (Universidad Complutense deMadrid), N. Shanmugalingam (University of Cincinnati), J. Tyson(University of Illinois at Urbana-Champaign), and A. Williams(Texas Tech University).

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The Bishop-Phelps-Bollobas TheoremDomingo Garcıa (Universidad de Valencia — Spain)

We prove versions of the Bishop-Phelps-Bollobas Theorem for operators, bi-linear forms and n-homogeneous polynomials on Banach spaces. We givea necessary condition on a Banach space Y in order that a (continuous)bilinear form on L1(µ) × Y that almost attains its norm at a couple ofelements (x0, y0) can be approximated by a bilinear form that attains itsnorm at a couple of elements close to (x0, y0). In case that Y is an Asplundspace we characterize the Banach spaces Y satisfying the previous prop-erty. As a consequence, we provide classes of Banach spaces Y for whicha version of Bishop-Phelps-Bollobas Theorem is satisfied for bilinear formson L1(µ)× Y . If X is a uniformly convex Banach space, then this theoremholds for the space P(nX;Y ) of all n-homogeneous polynomials from X intoan arbitrary Banach space Y .

Joint work with Acosta, Aron, Becerra, Choi, Kim, Lee, andMaestre.

Free spaces over compact metric spacesGilles Godefroy (Universite Paris VI — France)

The natural predual of the space of Lipschitz functions on a metric spaceM is called the Lipschitz-free Banch space over M . In a joint work with N.Ozawa, we show that even when the metric space M is very good (e.g., iscompact), the corresponding free space can be rather bad, e.g., it can failthe approximation property. Several consequences follow, in particular onextensions of real-valued Lipschitz maps from a subset G of a finite metricspace H to the whole set H.

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Derivatives of bivariate polynomials, Markov’stheorem and Geronimus nodesLawrence A. Harris (University of Kentucky — USA)

An outstanding problem that has been recently solved is to prove V. A.Markov’s theorem for derivatives of polynomials on any real normed linearspace. An elementary argument leads to a reduction of the problem to acertain directional derivative on two dimensional spaces. To state this, letPm(R2) denote the space of all polynomials p(s, t) of degree at most m andlet

Nk = {(cos(nπ/m), cos(qπ/m)) : n− q = k mod 2, 0 ≤ n, q ≤ m}.Then to prove the Markov theorem it suffices to show that the maximumof the values |Dkp(1, 1)(1,−1)| over polynomials p in Pm(R2) satisfying|p(x)| ≤ 1 for all x in the set Nk of nodes is attained when p(s, t) = Tm(s),where Tm is the Chebyshev polynomial of degree m.

We consider more general sets of nodes, called Geronimus nodes, wherethe extremal polynomials sought are orthogonal polynomials satisfying athree-term recurrence relation with constant coefficients. For example, thisincludes the Chebyshev polynomials of kinds 1–4.

In the course of our discussion we obtain an explicit formula for Lagrangepolynomials and a Lagrange interpolation theorem for the Geronimus nodes.We also deduce a bivariate cubature formula analogous to Gaussian quad-rature.

References

[1] J. Geronimus, On a set of polynomials, Ann. of Math. 31 (1930), 681–686.

[2] L. A. Harris, A proof of Markov’s theorem for polynomials on Banach spaces, J.Math. Anal. Appl. 368 (2010), 374–381.

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Mapping properties of the discrete maximal operatorin metric measure spacesJuha Kinnunen (Aalto University — Finland)

In this talk we consider the definition and mapping properties of so-calleddiscrete maximal operator in a metric measure space equipped with a dou-bling measure and a Poincare type inequality. The definition of the discretemaximal function is based on approximations of the function with Lipschitzpartition of unities in different scales. The obtained maximal function isequivalent with the standard Hardy-Littlewood maximal function with two-sided inequalities, but it seems to have better regularity properties. Wediscuss the behavior of the discrete maximal operator in Holder, Sobolev,Morrey and Campanato spaces.

The talk is based on joint works with D. Aalto, T. Heikkinen,V. Latvala, and H. Tuominen.

Non linear quotients and asymptotic uniform structureof Banach spacesGilles Lancien (Universite de Franche-Comte — France)

Recently, Lima and Randrianarivony pointed out the role of the property(β) of Rolewicz in nonlinear quotient problems, and answered a ten-year-old question of Bates, Johnson, Lindenstrauss, Preiss and Schechtman. Inthis talk, we will explain how their technique can be used to compare themodulus of asymptotic uniform smoothness of the range space with the (β)-modulus of the domain space. We also provide conditions under which thiscomparison can be improved.

Joint work with S. Dillworth, D. Kutzarova, and N. Randria-narivony.

Weighted composition operators on Bloch typespaces: new estimates of the essential normMikael Lindstrom (University of Oulu — Finland)

We will discuss the essential norm of weighted composition operators uCϕacting on Bloch-type spaces Bα in terms of the analytic function u : D→ Cand the n-th power of the analytic selfmap ϕ of the open unit disc D. Weobtain new characterizations for boundedness and compactness of uCϕ :Bα → Bβ for all 0 < α, β <∞, thus answering an open question of Manhasand Zhao concerning the case α = 1.

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Monomial expansions of Hp-functions in infinitelymany variablesManuel Maestre (Universidad de Valencia — Spain)

In the late sixties Lumer, Hoffman, Rossy and Konig, developed a functiontheory of Hardy spaces in the general context of uniform algebras. In 1985Cole and Gamelin ([1]) considered the following natural case. They took inthe infinite torus TN the Haar measure measure m, realized as the countableproduct of copies of the normalized Lebesgue measure on the torus T; andthey studied the Hardy spaces Hp(TN) (0 < p ≤ ∞) defined as the elements

f of Lp(TN, dm) with their Fourier coefficients f(α) = 0 whenever the multi-

index α ∈ Z(N) \ N(N)0 .

In [4] we describe monHp(TN), the set of points z in the infinite polydisk

DN for which the formal series of monomials∑

α∈N(N) f(α)zα converge for

all f ∈ Hp(TN).To our approach is very relevant the isometry obtained in [2] between

Hp(TN) and the Banach space H∞ of Dirichlet series bounded in the righthalf plane, recent results on Dirichlet series ([3]), and our previous work onmonomial expansions ([5]).

Joint work with A. Defant, L. Frerick, and P. Sevilla.

References

[1] B. J. Cole and T. W. Gamelin, Representing measures and Hardy spaces for theinfinite polydisk algebra, Proc. London Math. Soc. (3) 53 (1986), no. 1, 112–142.

[2] H. Hedenmalm, P. Lindqvist, and K. Seip, A Hilbert space of Dirichlet series andsystems of dilated functions in L2(0, 1), Duke Math. J. 86 (1997), no. 1, 1–37.

[3] R. Balasubramanian, B. Calado, and H. Queffelec, The Bohr inequality forordinary Dirichlet series, Studia Math. 175 (2006), no. 3, 285–304.

[4] A. Defant, L. Frerick, M. Maestre, and P. Sevilla-Peris, Monomial expansionsof Hp–functions in infinitely many variables, preprint.

[5] A. Defant, M. Maestre, and C. Prengel, Domains of convergence for monomialexpansions of holomorphic functions in infinitely many variables, J. Reine Angew.Math. 634 (2009), 13–49.

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New advances and directions in the study ofabsolutely summing (linear and nonlinear) operatorsDaniel Pellegrino (Universidade Federal da Paraıba — Brazil)

In this talk we present an overview of results of several authors related toabsolutely summing linear and nonlinear operators. Among other results wepresent new versions of the Pietsch Domination Theorem, inclusion results,etc. We also discuss the effect of cotype in the nonlinear theory and thesearch of the “perfect” generalization of the concept of absolutely summingoperators to polynomials and multilinear mappings (from the point of viewof the theory of operator ideals).

References

[1] G. Botelho, H.-A. Braunss, H. Junek, and D. Pellegrino, Holomorphy typesand ideals of multilinear mappings, Studia Math. 177 (2006), 43–65.

[2] G. Botelho, C. Michels, and D. Pellegrino, Complex interpolation and summa-bility properties of multilinear operators, Rev. Matem. Complutense 23 (2010), 139–161.

[3] G. Botelho and D. Pellegrino, When every multilinear mapping is multiple sum-ming, Math. Nachr. 282 (2009), 1414–1422.

[4] D. Carando, V. Dimant, and S. Muro, Coherent sequences of polynomials idealson Banach spaces, Math. Nachr. 282 (2009), 1111–1133.

[5] A. Defant, L. Frerick, J. Ortega-Cerda, M. Ounaıes, and K. Seip, TheBohnenblust-Hille inequality for homogeneous polynomials is hypercontractive, Ann.of Math. (2) 174 (2011), 485–497.

[6] A. Defant, D. Popa, and U. Schwarting, Coordenatewise multiple summing op-erators on Banach spaces, J. Funct. Anal. 259 (2010), 220–242.

[7] J. Diestel, H. Jarchow, and A. Tonge, Absolutely summing operators, Cam-bridge University Press, 1995.

[8] M. C. Matos, Fully absolutely summing and Hilbert-Schmidt multilinear mappings,Collectanea Math. 54 (2003), 111–136.

[9] M. C. Matos, Nonlinear absolutely summing mappings, Math. Nachr. 258 (2003),71–89.

[10] D. Pellegrino and J. Santos, Absolutely summing operators: A panorama,Quaest. Math. 34 (2011), 447–478.

[11] , On summability of nonlinear mappings: A new approach, Math. Z. 229(2012), 189–196.

[12] D. Pellegrino, J. Santos, and J. B. Seoane-Sepulveda, Some techniques onnonlinear analysis and applications, Adv. in Math. 229 (2012), 1235–1265.

[13] D. Perez-Garcıa, Operadores multilineales absolutamente sumantes, Thesis, Uni-versidad Complutense de Madrid, 2003.

[14] , The inclusion theorem for multiple summing operators, Studia Math. 165(2004), 275–290.

[15] A. Pietsch, Ideals of multilinear functionals, Proceedings of the Second Interna-tional Conference on Operator Algebras, Ideals and Their Applications in TheoreticalPhysics, 185–199, Teubner-Texte, Leipzig, 1983.

[16] D. Popa, Reverse inclusions for multiple summing operators, J. Math. Anal. Appl.350 (2009), 360–368.

[17] D. Puglisi and J. B. Seoane-Sepulveda, Bounded linear non-absolutely summingoperators, J. Math. Anal. Appl. 338 (2008), 292–298.

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Functional analysis and geometry of convex cones inBanach spacesAlberto Seeger (Universite d’Avignon — France)

This talk, divided into three parts, explains how various topological and geo-metric properties of normed spaces (completeness, reflexivity, smoothness,rotundity, etc) have a bearing in the analysis of convex cones.

• Part I reviews several ways of measuring the distance between apair of closed convex cones in a normed space (X, ‖ · ‖). We showhow the “sphericity coefficient” of X intervenes in a natural waywhen it comes to compare the gap distance and the truncatedPompeiu-Hausdorff distance.• Part II deals with various geometric properties of a closed convex

cone (solidity, reproducibility, etc). The nature of the underlyingnormed space (X, ‖ · ‖) plays a crucial role in the discussion.• Part III addresses the issue of formalizing the concept of center of a

closed convex cone. Four distinct proposals are studied in detail:the incenter, the circumcenter, the inner center, and the outercenter.

Parts I and II are based on joint work with Alfredo Iusem(IMPA, Rio de Janeiro) and Part III is based on joint work withRene Henrion (WIAS, Berlin).

SHORT TALKS

17

A continuous image of a Radon-Nikodym compactspace which is not Radon-NikodymAntonio Aviles (Universidad de Murcia — Spain)

We construct a continuous image of a Radon-Nikodym compact space whichis not Radon-Nikodym compact, solving the problem posed in the 80ties byIsaac Namioka.

Joint work with Piotr Koszmider (Polish Academy of Sciences).

Nicodemi sequences of operators between spaces ofmultilinear mappingsGeraldo Botelho (Universidade Federal de Uberlandia — Brazil)

We obtain results on three aspects of Nicodemi extensions of multilinearmappings between Banach spaces: (i) subspace invariance, (ii) the normsof the extension operators, (iii) when Aron-Berner extensions are Nicodemiextensions.

Joint work with Kuo Po Ling (Universidade Federal deUberlandia).

Polynomials and positive tensor productsGerard J. Buskes (University of Mississippi — USA)

In this talk we introduce several quotients of the Fremlin tensor product ofvector lattices and Banach lattices to study homogeneous polynomials. Wewill also connect these quotients to concavifications of Banach lattices.

Joint work with Qingying Bu, Vladimir Troitsky, Alexey Popov,and Adi Tcaciuc.

18

Devaney chaos and distributional chaos in thesolutions of certain partial differential equationsJ. Alberto Conejero (Universitat Politecnica de Valencia — Spain)

The notion of distributional chaos has been rencently added to the study ofthe linear dynamics of operators and C0-semigroups of operators.

A criterion for distributional chaos and the existence of a dense distribu-tionally irregular manifold for a C0-semigroup has been recently obtainedin [1].

We apply it to several examples of C0-semigroups that were already knownto be chaotic in the sense of Devaney, some of them by applying the Desch-Schappacher-Webb Criterion. These results motivate us to wonder where isthe limit between both notions of chaos.

Joint work with Xavier Barrachina (Universitat Politecnica deValencia).

References

[1] A. A. Albanese, X. Barrachina, E. M. Mangino, and A. Peris. Distributionallyirregular vectors (2012), preprint.

[2] X. Barrachina and J. A. Conejero, Devaney chaos and distributional chaos in thesolutions of certain partial differential equations (2012), preprint.

[3] W. Desch, W. Schappacher, and G. F. Webb, Hypercyclic and chaotic semigroupsof linear operators, Ergodic Theory Dynam. Systems 17 (1997), no. 4, 793–819.

[4] S. El Mourchid, The imaginary point spectrum and hypercyclicity, Semigroup Forum73 (2006), no. 2, 313–316.

Uniform Eberlein compacta and coincidence of Pettisand McShane integrabilityMarian Fabian (Czech Academy of Sciences — Czech Republic)

McShane integral is a close relative of Henstock-Kurzweil integral. Let Kbe a compact space and let f : [0, 1]→ C(K) be a function. If f is McShaneintegrable, it is automatically Pettis integrable. We focus on conditions onK under which the converse holds. If K is a uniform Eberlein compact, thenthe both integral coincide (Deville, Reodriguez) while there exist an Eberleincompact K and a scalarly null (hence Pettis integrable) f : [0, 1] → C(K)which is not McShane integrable (Aviles, Plebanek, Rodriguez). We tryto study/diminish the slot in between these two results. A central conceptbehind this problem is the so called MC-filling of a family of sets. Several(counter)examples in this direction are presented and discussed.

19

Maximal spaceability in topological vector spacesVinicius Favaro (Universidade Federal de Uberlandia — Brazil)

In this paper we introduce a new technique to prove the existence of closedsubspaces of maximal dimension inside sets of topological vector sequencespaces. The results we prove cover some sequence spaces not studied beforein the context of spaceability and settle some questions on classical sequencespaces that remained open.

Joint work with Geraldo Botelho (Universidade Federal deUberlandia), Daniel Cariello (Universidade Federal deUberlandia), and Daniel Pellegrino (Universidade Federal daParaıba).

Geometry of integral polynomials, M-ideals andunique norm preserving extensionsDaniel Eric Galicer (Universidad de Buenos Aires — Argentina)

We use the Aron-Berner extension to prove that the set of extreme points ofthe unit ball of the space of integral k-homogeneous polynomials over a realBanach space X is {±φk : φ ∈ X∗, ‖φ‖ = 1}. With this description we showthat, for real Banach spaces X and Y , if X is a non trivial M -ideal in Y , then⊗k,s

εk,sX (the k-th symmetric tensor product of X endowed with the injective

symmetric tensor norm) is never an M -ideal in⊗k,s

εk,sY . This result marks

up a difference with the behavior of non-symmetric tensors since, when X

is an M -ideal in Y , it is known that⊗k

εkX (the k-th tensor product of X

endowed with the injective tensor norm) is an M -ideal in⊗k

εkY .

Nevertheless, if X is also Asplund, we prove that every integral k-homo-geneous polynomial in X has a unique extension to Y that preserves theintegral norm.

Other applications to the metric and isomorphic theory of symmetric ten-sor products and polynomial ideals are also given.

Joint work with Veronica Dimant (Universidad de San Andres)and Ricardo Garcıa (Universidad de Extremadura).

20

Algebrability and strong algebrabilitySzymon G lab (Technical University of Lodz — Poland)

We define a strong algebrability which is a stronger version of the notionof algebrability. We prove several new facts and we strengthen old ones onalgebrability.

Metric currents in the planeJasun Gong (Aalto University (TKK) — Finland)

In 2000 Ambrosio and Kirchheim developed a theory of currents in the set-ting of complete metric spaces. As a consequence, they showed that muchof the geometric measure theory on Rn extends to more general settings,including a large class of Banach spaces. These results include the isoperi-metric inequality and more generally, the Plateau problem.

Despite the utility of the Ambrosio-Kirchheim theory, there is no precisecharacterization of such objects in general. On Euclidean spaces, it is knownthat “metric currents” are indeed currents in the usual sense, but it remainsa conjecture that such objects also satisfy Whitney’s flatness criterion.

In this talk we discuss fine properties of metric currents and prove theconjecture in the case of the plane.

BPB property for numerical radius in `1(C)Antonio Jose Guirao S. (Universidad Politecnica de Valencia —Spain)

We provide two constructive versions for `1(C) of the classical Bishop-Phelps-Bollobas theorem and, as an application, we show that the laterspace and c0(C) satisfy Bishop-Phelps-Bollobas property for the numericalradius.

Joint work with Olena Kozhushkina (Kent State University).

21

Weighted Banach spaces of harmonic functionsEnrique Jorda (Universidad Politecnica de Valencia — Spain)

Weighted Banach spaces of holomorphic and harmonic functions have beenrecently studied by several authors as Bierstedt, Bonet, Boyd, Domanski,Lindstrom, Lusky, Rueda, Taskinen and Wolf. In this work we introducethe associated weight vh associated to the space of harmonic functions. Weextend results of Boyd and Rueda giving conditions for the equality v = vhand we extend under these conditions results about of the essential normof a composition operator between spaces of holomorphic functions due toBonet, Domanski, Lindstrom and Montes to composition operators definedon weighted Banach spaces of pluriharmonic functions with holomorphicsymbol. We also prove that for a domain G ⊆ Rn the weighted spacehv0(G) formed by functions vanishing at infinity on G is isomorphic to aclosed subspace of c0, extending then previous work of Bonet and Wolf andKalton and Werner.

Joint work with Ana Marıa Zarco (Universidad Politecnica de Va-lencia).

22

Spaceability of sets of nowhere Lq functionsPedro Kaufmann (Universidade de Sao Paulo — Brazil)

Let µ be a Borel measure on a Polish space X. We say that a functionf : X → R is nowhere Lq if, for each nonvoid open subset U of X, therestriction f |U is not in Lq(U, µ).

For a fixed 0 ≤ p <∞, let us denote

Sp.= {f ∈ Lp(µ) : f is nowhere Lq, for each p < q ≤ ∞},

and

S′p.= Sp \ ∪0<q<pLq(µ).

We shall show the following:

(1) If µ is atomless and µ(U) > 0 for each nonepty open subset U ⊂ X,then Sp ∪ {0} admits an isometric copy of `p;

(2) if µ is infinite and σ-finite, then Lp(µ) \⋃

0<q<p Lq(µ) admits an

isometric copy of `p;(3) if µ satisfies the conditions in (1) and (2), then S′p ∪ {0} admits an

isometric copy of `p.

In all of the above three cases, the copy of `p is complemented in Lp, underthe assumption that p ≥ 1. This brings some generalizations and improve-ments to recent spaceability results by Botelho, Diniz, Favaro, Pellegrinoand Seoane-Sepulveda.

Joint work with Szymon G lab (Technical University of Lodz) andLeonardo Pellegrini (Universidade de Sao Paulo).

Hardy-Littlewood inequalities for norms of positiveoperators on sequence spacesMiguel Lacruz Martın (Universidad de Sevilla — Spain)

We consider some estimates of Hardy and Littlewood for norms of operatorson sequence spaces, and we exploit a factorization result of Maurey to obtainimproved bounds and simplified proofs for the case of a positive operator.

23

Smooth renormings of the Lebesgue-Bochner functionspace L1(µ,X)Sebastian Lajara (Universidad de Castilla-La Mancha — Spain)

We show that, if µ is a probability measure and X is a Banach space,then the space L1(µ,X) of Bochner integrable functions admits an equiva-lent Gateaux (uniformly Gateaux) smooth norm provided that X has sucha norm, and that if X admits an equivalent Frechet (uniformly Frechet)smooth norm, then L1(µ,X) has an equivalent renorming whose restrictionto every reflexive subspace is Frechet (uniformly Frechet) smooth.

Joint work with Marian Fabian (Institute of Mathematics, CzechAcademy of Sciences).

Lineability of the set of holomorphic mappings withdense rangeJeronimo Lopez-Salazar Codes (Universidad Complutense de Madrid— Spain)

Let D be the open unit disc in C. Let E be a separable Banach space and letBE be the open unit ball of E. At the Conference on Infinite DimensionalHolomorphy held at the University of Kentucky in 1973, D. Patil askedwhether there exist holomorphic mappings f : D → BE such that f (D)is dense in BE . In 1976, R. M. Aron obtained a positive answer to thisquestion. At the same time, J. Globevnik and W. Rudin independentlyproved that the result also holds if the ball BE is replaced by any connectedopen subset U of E. We will study the set

{f ∈ H (D, U) : f (D) is dense in U}of holomorphic mappings with dense range. We will prove the lineabilityand density of this set for different choices of U ⊂ E.

Joint work with Richard M. Aron (Kent State University).

24

The specification property in the dynamics of linearoperatorsFelix Martınez Jimenez (Universidad Politecnica de Valencia —Spain)

We introduce the notion of the Specification Property (SP) for operators onBanach spaces, inspired by the usual one of Bowen for continuous maps oncompact spaces. This is a very strong dynamical property related to thechaotic behaviour. Several general properties of operators with the SP areestablished. For instance, every operator with the SP is mixing, Devaneychaotic, and frequently hypercyclic. In the context of weighted backwardshifts, the SP is equivalent to Devaney chaos. In contrast, there are Devaneychaotic operators (respectively, mixing and frequently hypercyclic operators)which do not have the SP.

Joint work with Alfredo Peris (Universidad Politecnica de Valen-cia).

25

On the polynomial Lindenstrauss theoremMartın Mazzitelli (Universidad de Buenos Aires — Argentina)

The Bishop-Phelps theorem [1] states that for any Banach space X, theset of norm attaining linear bounded functionals is dense in X ′, the dualspace of X. Since then, the study of norm attaining functions has attractedthe attention of many authors. Lindenstrauss showed in [2] that there isno Bishop-Phelps theorem for linear bounded operators. Nevertheless, heproved that the set of bounded linear operators (between any two Banachspaces X and Y ) whose second adjoints attain their norm, is dense in thespace of all operators. This result was later extended by Acosta, Garcıa andMaestre [3] for multilinear operators.

In the context of homogeneous polynomials, where symmetry representsan additional difficulty, Aron, Garcıa and Maestre showed in [4] the densityof the scalar valued 2-homogeneous polynomials whose Aron-Berner exten-sion attain their norm.

Under certain hypothesis on the space X, we show that a Lindenstrausstheorem holds for N -homogeneous polynomials from X into any dual space(and, therefore, for scalar-valued polynomials on X). For this, we presentan integral representation for the elements of some tensor products.

We also exhibit many situations in which there is no polynomial Bishop-Phelps theorem but our results apply. In particular, we present couples ofBanach spaces which do not satisfy the polynomial Bishop-Phelps theoremfor any degree N ≥ 1 but satisfy the polynomial Lindenstrauss theorem forevery degree.

Joint work with Daniel Carando and Silvia Lassalle (Universidadde Buenos Aires).

References

[1] E. Bishop and R. Phelps, A proof that every Banach space is subreflexive, Bull.Amer. Math. Soc. 67 (1961), 97–98.

[2] J. Lindenstrauss, On operators which attain their norm, Israel J. Math. 1 (1963),139–148.

[3] M. Acosta, D. Garcıa, and M. Maestre, A multilinear Lindenstrauss theorem, J.Funct. Anal. 235 (2006), no. 1, 122–136.

[4] R. Aron, D. Garcıa, and M. Maestre, On norm attaining polynomials, Publ. Res.Inst. Math. Sci. 39 (2003), no. 1, 165–172.

[5] M. Jimenez Sevilla and R. Paya, Norm attaining multilinear forms and polynomialson preduals of Lorentz sequence spaces, Studia Math. 127 (1998), no. 2, 99–112.

26

Hypercyclic composition operators on H0v -spaces

Alejandro Miralles (Universitat Jaume I — Spain)

We consider analytic self-maps φ on D and prove that the composition oper-ator Cφ acting on H0

v is hypercyclic if φ is an automorphism or a hyperbolicnon-automorphic symbol with no fixed point. We give examples of weightsv and parabolic non-automorphisms φ on D which yield non-hypercycliccomposition operators Cφ on H0

v .

Joint work with Elke Wolf (University of Paderborn).

27

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

If E is a complex Banach algebra, a mapping f : U ⊂ E → E is Lorchanalytic if given any a ∈ U there exists ρ > 0 and there exist unique elementsan ∈ E, such that f(z) =

∑∞n=0 an(z − a)n, for all z in ‖z − a‖ < ρ. The

theory of Lorch-analytic mappings goes back to the 1940’s and is a verynatural extension of the classical concept of analytic function to infinitedimensional algebras that allows concepts as Laurent series, singularities ora Mittag-Leffler’s theorem (see [1] and [2]). In this talk we are going to studytopological and algebraic properties of algebras of analytic mappings (in thesense of Lorch) in connection with the topological and algebraic propertiesof the underlying spaceE.

In conection we consider the space

Γ(E) ={

(an)n ⊂ E ; limn→∞

‖an‖1n = 0

}.

Endowed with the usual operations of adition, product by scalar and prod-uct, Γ(E) is an algebra. This algebra, endowed with the topology associatedto the metric

d(a, b) = sup{‖a0 − b0‖; ‖an − bn‖1n , n ∈ N}

is algebraically isomorphic to the algebra of the mappings from E into Ethat are analytic in the sense of Lorch (under the Hadamard product) andthe study of this algebra leads to a better knowledge of the topological andalgebraic properties of Γ(E).

Joint work with Alex F. Pereira (Universidade Federal do Rio deJaneiro).

References

[1] E. K. Blum, A theory of analytic functions in Banach algebras, Trans. Amer. Math.Soc. 78 (1955) 343–370.

[2] B. W. Glickfeld, Meromorphic functions of elements of a commutative Banach al-gebra, Trans. Amer. Math. Soc. 151 (1970), no. 1, 293–307.

28

Algebras of bounded type functions associated to aholomorphy typeSantiago Muro (Universidad de Buenos Aires — Argentina)

We study spaces of bounded type holomorphic functions associated to agiven holomorphy type. We show that in several cases this spaces are locallym-convex algebras. We prove that the spectrum has a natural analyticstructure, which we use to characterize the envelope of holomorphy. We alsostudy whether the extension to the envelope of holomorphy of a function ofa given type is also of the same type, a question made by Hirschowitz in [1].

References

[1] A. Hirschowitz, Prolongement analytique en dimension infinie, Ann. Inst. Fourier,22 (1972), no. 2, 255–292.

Diameter 2 properties in Banach spacesOlev Nygaard (University of Agder — Norway)

If X is a uniform algebra, if X has the Daugavet property or if X is M -embedded, every relatively weakly open subset of the unit ball of the Banachspace X is known to have diameter 2, i.e., X has the diameter 2 property.Note that in particular every slice will have diameter 2, so these spaces are”opposite” to the spaces with the Radon-Nikodym property.We explain that there is in fact three types of diameter 2 properties, oneformally weaker and one formally stronger then the one above. Then weshow that the above three classes of spaces do not contain all diameter 2spaces and we discuss whether these three classes also enjoy the strongestdiameter 2 property.

Joint work with Trond Abrahamsen (University of Agder) andVegard Lima (University of Alesund).

29

Lower bounds for norms of products of polynomialson `pDamian Pinasco (Universidad Torcuato Di Tella — Argentina)

Let P1, . . . , Pn be continuous polynomials defined on a complex Banachspace E, and suppose that we have a norm ‖ · ‖ defined on the space ofpolynomials over E. The problem of finding a constant M , depending onlyon the degrees of P1, . . . , Pn, such that

(1) ‖P1‖ · · · ‖Pn‖ ≤M ‖P1 · · ·Pn‖was studied by many authors. For the uniform norm ‖P‖ =sup‖x‖E=1 |P (x)|, C. Benıtez, Y. Sarantopoulos and A. Tonge [1], proved

that if Pi has degree ki, for 1 ≤ i ≤ n, then (1) holds with constant

M =(k1 + · · ·+ kn)(k1+···+kn)

kk11 · · · kknn

for any complex Banach space. The authors also showed an example on `1for which the equality prevails. However, for many spaces it is possible toimprove this constant. For instance, in [3], we showed that for a complexHilbert space H, if Pi is a ki-homogeneous polynomial for 1 ≤ i ≤ n andn ≤ dim(H), then

M =

√(k1 + · · ·+ kn)(k1+···+kn)

kk11 · · · kknn

is optimum. We will show that inequality (1) holds for E = `p (1 < p ≤ 2),if we consider

M =

((k1 + · · ·+ kn)(k1+···+kn)

kk11 · · · kknn

)1/p

.

Also, this bound is sharp.

Joint work with Daniel Carando (Universidad de Buenos Aires)and Jorge T. Rodrıguez (Consejo Nacional de InvestigacionesCientıficas y Tecnicas).

References

[1] C. Benıtez, Y. Sarantopoulos, and A. Tonge, Lower bounds for norms of prod-ucts of polynomials, Math. Proc. Cambridge Philos. Soc. 124 (1998), no. 3, 395–408.

[2] D. Carando, D. Pinasco, and J. T. Rodrıguez, Lower bounds for norms of prod-ucts of polynomials on `p, preprint.

[3] D. Pinasco, Lower bounds for norms of products of polynomials via Bombieri inequal-ity, Trans. Amer. Math. Soc., to appear.

30

Measurability in C(2κ) and Kunen cardinalsJose Rodrıguez (Universidad de Murcia — Spain)

A cardinal κ is called a Kunen cardinal if the σ-algebra on κ×κ generated byall products A×B, where A,B ⊆ κ, coincides with the power set of κ×κ. Forany cardinal κ, let C(2κ) be the Banach space of all continuous real-valuedfunctions on the Cantor cube 2κ. We prove that κ is a Kunen cardinal if andonly if the Baire σ-algebra on C(2κ) for the pointwise convergence topologycoincides with the Borel σ-algebra on C(2κ) for the norm topology. Someother links between Kunen cardinals and measurability in Banach spacesare also given.

Joint work with A. Aviles and G. Plebanek.

Projective tensor products of Lp-spacesPilar Rueda (Universidad de Valencia — Spain)

It is well-known that the space of continuous n−homogeneous polynomialsdefined on a Banach space X has as a predual the symmetric s−projectivetensor product ⊗n,sπs X. We prove that, whenever X = Lp(µ) for some regular

probability measure µ on a compact Hausdorff space, the predual ⊗n,sπs Lp(µ)can be described explicitely as the completion of a subspace of Lp/n(µ) withrespect to an explicitely constructed norm.

Joint work with Geraldo Botelho (Universidade Federal deUberlandia) and Daniel Pellegrino (Universidade Federal daParaıba).

Reproducing kernel Banach spaces and learningmachinesLuis Sanchez Gonzalez (Universidad Complutense de Madrid —Spain)

We develop a theory of reproducing kernel to Banach spaces and apply ourconstruction to the basis learning algorithms, in particular, to the supportvector machines.

Joint work with Pando Georgiev and DaPanos Pardalos (Univer-sity of Florida).

31

A general version of the Pietsch Domination TheoremJoedson Santos (Universidade Federal de Sergipe — Brazil)

We present a characterization of the arbitrary nonlinear mappings f : X1 ×· · · × Xn → Y between Banach spaces that satisfy a quite natural PietschDomination-type theorem around a given point (a1, . . . , an) ∈ X1×· · ·×Xn.More generally, we prove a general theorem which contains, as particularcases, several previous versions of Pietsch-type theorems, including a recentunified version of the Pietsch Domination Theorem due to G. Botelho, D.Pellegrino, and P. Rueda.

Joint work with Daniel Pellegrino (Universidade Federal daParaıba) and Juan B. Seoane-Sepulveda (UniversidadComplutense de Madrid).

References

[1] G. Botelho, D. Pellegrino, and P. Rueda, A unified Pietsch DominationTheorem, Journal of Mathematical Analysis and Applications 365 (2010), 269–276.

[2] D. Pellegrino and J. Santos, On summability of nonlinear mappings: a newapproach, to appear in Mathematische Zeitschrift.

[3] D. Pellegrino, J. Santos, and J. B. Seoane-Sepulveda, Some techniques onnonlinear analysis and applications, Advances in Mathematics, doi:10.1016/j.aim.2011.09.014.

[4] A. Pietsch, Absolut p-summierende Abbildungen in normieten Raumen, StudiaMathematica 27 (1967), 333–353.

Polarization constants and the plank problemYannis Sarantopoulos (National Technical University — Greece)

We give new results and we discuss some open questions related to polar-ization constants of polynomials on Banach spaces and the plank problemof Tarski.

Differences of composition operators betweenBergman spacesErno Saukko (University of Oulu — Finland)

An interesting connection between differences of composition operators andcertain weighted composition operators is discussed. Furthermore, charac-terizations for boundedness and compactness of differences of compositionoperators are presented in terms of these weighted composition operators inthe Bergman space setting.

32

Spectral properties of disjointly strictly singularoperatorsPedro Tradacete (Universidad Carlos III de Madrid — Spain)

Spectral properties of strictly singular and disjointly strictly singular op-erators on Banach lattices are studied. We show that even in the case ofpositive operators, the whole spectral theory of strictly singular operatorscannot be extended to disjointly strictly singular. However, several spectralproperties of disjointly strictly singular operators are given.

On p-compact polynomials and holomorphic mappingsPablo Turco (Consejo Nacional de Investigaciones Cientıficas yTecnicas — Argentina)

Inspired in Grothendieck’s result which characterize relatively compact setsas those contained in the convex hull of a norm null sequence of vectors of thespace, Sinha and Karn introduced the concept of relatively p-compact sets.Loosely speaking, these sets are determined by norm p-summable sequences.Associated with relative p-compact sets, we have naturally defined the notionof p-compact mappings.

The aim of this talk is to show some results concerning p-compact poly-nomials and p-compact holomorphic mappings. We characterize p-compactholomorphic functions in terms of its Taylor series expansion and an appro-priate radius of convergence. As a consequence we present some examplesshowing that p-compact holomorphic functions behave more like nuclearthan compact mappings.

Joint work with Silvia Lassalle (Universidad de Buenos Aires).

33

Contractive projections on Banach spacesDavid Yost (University of Ballarat — Australia)

On all of the familiar Banach spaces, it is very easy to write down a formulawhich defines a norm one projection. This is not true for arbitrary Banachspaces. The following rough trichotomy indicates the range of behaviour.

A) Most finite dimensional Banach spaces admit no norm one projec-tions at all (except those with rank one). Such examples may besmooth, strictly convex, or polyhedral.

B) Some separable Banach spaces admit no continuous projections atall (except those with finite rank or co-rank).

C) Many non-separable Banach spaces admit many nontrivial norm oneprojections, even under every equivalent norm.

Joint work with Jerzy Grzybowski (Adam Mickiewicz University).