BOOK OF ABSTRACTS

Function Theory on Infinite Dimensional Spaces VIII

Madrid, 15-18 December 2003.

Index of Abstracts

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

Plenary talks

R. M. Aron Properties of certain sets of functions having special

properties 3

Y. Benyamini Nonexpansive Metric Projections in Spaces of Continuous

Functions 3

R. Deville On the range of the derivative of a Gateaux-smooth mapping

between Banach spaces 3

M. Fabian Weakly compactly generated spaces and their relatives 4 T. Gamelin Composition Operators on Uniform Algebras 4 G. Godefroy Some Recent Results on Hypercyclic Operators 4 B. Grecu Schauder Bases for Symmetric Tensor Products 5 L. Harris Holomorphic Functions on Domains in Operator Spaces 5 J. A. Jaramillo Polynomial Topologies on a Banach Space 5 R. Pérez-Marco KAM theory in Infinite Dimensional Spaces 5 R. Ryan KAM Polynomials on Banach Spaces with Unconditional

Schauder Bases 6

I. Zalduendo Integral Representation of Holomorphic Functions on Banach Spaces 6

Short talks

J. M. Almira. On the Müntz Theorem for numerable compact sets 8 A. Avilés Radon-Nikodým compact spaces and Banach spaces of low

weight 8

A. Bayoumi Foundations of complex Analysis in non locally Convex

Spaces 9

J. Bes Operators with common Hypercyclic subspaces 9 E. Corbacho Automatic continuity of directional derivatives 9 S. Delpech Approximation of uniformly continuous mappings in Orlicz

spaces 10

J.Ferrera Viscosity solutions to Hamilton-Jacobi equations on

Riemannian manifolds 11

C. Finet Composition operators on a Hilbert space of Dirichlet Series 11 R. Fry Smooth Approximation on Banach Spaces 11 P. Galindo Polynomials generated by linear operators 12 T. Gaspari Optimal Lipschitz Extensions 12 A. Kitover Peripheral spectrum of positive operators and

generalizations of Jentzsch’s theorem 12

S. Lajara Average locally uniformly rotund dual norms 13 S. Lassalle Canonical isometries between spaces of polynomials 13 M. Lindström Spectra of weighted composition operators on weighted

Banach spaces of analytic functions 13 M. Matolcsi Linear polarization constant of ℝ^{n} 14 J. Merí Finite-Dimensional Banach Spaces with numerical index zero 15 A. Montes The Volterra operator is not supercyclic 15 A. Montesinos Deleting diffeomorphisms and starlike bodies

in Banach spaces 16

V. Montesinos Compacta in product of lines from a functional-analytic

point of view 16

Y. Moreno Vector valued Sobczyk’s theorem 17 N. Palmberg Spectra of composition operators on BMOA 18 D. Pérez Absolutely Summing Multilinear Operators 18 L. M. Pozo Hamiltonian structure of the harmonic functions on an

annulus 18

L. Oncina Kuratowski’s index of non-compactness and renorming

in Banach spaces 17

L. Quarta On lineability of sets of unimaximal continuous functions 19 S. Revesz Polarization constants, Plank problems and local Theory

in Banach spaces 20

J. Rodríguez The Birkhoff integral 21 V. M. Sánchez Strictly singular inclusions between symmetric

sequence spaces 21

A. Santos Banach Spaces and Polytopes 22 J. B. Seoane Construction of Infinite Dimensional Vector Spaces of

Hypercyclic Vectors 22

R. Vidal Strong minimality of Gaussian integrals 22 I. Villanueva Where do homogeneous polynomials on ℓ_{1}^{n} attain

their norm? 23

PLENARY TALKS

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Properties of certain sets of functions having special properties

Richard M. Aron. Kent State University (USA)

Abstract. We discuss work with Vladimir Gurariy and Juan Seoane, in which it is shown that the set of everywhere differentiable, nowhere monotone functions f : R → R contains an infinite dimensional vector space. We also hope to discuss recent work (by others) involving the set of entire functions f : C → C such that the set {f(z), f(z + a), …, f(z + na), …} is dense in H(C).

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Nonexpansive Metric Projections in Spaces of Continuous Functions

Yoav Benyamini. Technion-Institute of Technology (Israel)

Joint work with Rafael Espínola and Genaro López

Abstract. Let K be a subset of a Banach space E. We discuss the existence of a retraction φ : E → K which is simultaneously nonexpansive and a nearest point map. (We shall then say that K is a proximinal nonexpansive retract of E.)

Among other results we characterize subspaces of codimension one in spaces C(S) of continuous functions which are proximinal nonexpansive retracts of C(S) as well as the proximinal nonexpansive retracts in l_{∞}^{n}.

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On the range of the derivative of a Gateaux-smooth mapping between Banach spaces

Robert Deville. Université de Bourdeaux (France)

Abstract. We consider the following problem : Given two separable real Banach spaces X and Y, does there exist a Lipschitz continuous and bounded mapping from X to Y such that f is Gateaux differentiable at each point of X, and, for each x, y ∈ X we have ||f′(x) − f′(y)|| > 1 whenever x is different from y ? We show that this is not always possible (for instance this is not possible if dim(Y) = 1 ), but there are Banach spaces X and Y where this construction can be performed, for instance when X = Y is the separable Hilbert space, or when X = ℓ^{1} and dim(Y) = 2. This answers a question of Gilles Godefroy raised during the seventh edition of the International Conference on Infinite Dimensional Analysis held in Madrid in 2001.

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Weakly compactly generated spaces and their relatives

Marian Fabian. Czech Academy of Sciences (Czech Republic)

Abstract. A Banach space is called WCG it contains a weakly compact linearly dense subset. For instance, reflexive spaces or C(K), with K Eberlein compact, are such. The WCG spaces form an important subclass of nonseparable Banach spaces, with many nice properties. Their relatives are subspaces of WCG spaces, Vašák (i.e. WCD) spaces, weakly Lindelöf determined spaces, Hilbert generated spaces, subspaces of Hilbert generated spaces, ... We shall characterize each of these classes in terms of the presence of a total set therein with some extra properties. Other characterizations can be expressed via an equivalent norm with some kind of differentiability. A sample result: X is a subspace of a WCG space if and only if it admits a linearly dense set Γ ⊂ X and subsets Γ_{n}^{ε} ⊂ Γ, ε > 0, n ∈ IN, such that ⋃_{n∈IN} Γ_{n}^{ε} = Γ and for every x^{∗} ∈ X^{∗}, every ε > 0, and every n ∈ IN the set {γ ∈ Γ_{n}^{ε}; x^{∗}(γ) > ε} is finite. As a consequence we get a functional-analytic proof of a well known fact that a continuous image of a (uniform) Eberlein compact is such. Our technique also reproves Farmaki’s result on sitting (uniform) Eberlein compacta in Σ(Γ).

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Composition Operators on Uniform Algebras

Theodore Gamelin. UCLA (USA)

Abstract. The talk will focus on composition operators on uniform algebras and the existence of attracting cycles for the corresponding self-map of the spectrum of the uniform algebra. The pseudohyperbolic metric on the spectrum and hyperbolically bounded sets will play a decisive role.

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Some Recent Results on Hypercyclic Operators

Gilles Godefroy. Université Pais VI (France)

Abstract. Not supplied ——————————————————————————————

Schauder Bases for Symmetric Tensor Products

Bogdan Grecu. National University of Ireland (Ireland)

Abstract. For a Banach space E with Schauder basis, we prove that the n fold symmetric tensor product ˆ⊗_{μ, s}^{n}E has a Schauder basis for all symmetric uniform crossnorms μ. This is done by modifying the square ordering on ℕ^{n} and showing that the new ordering gives tensor product bases in both ˆ⊗_{μ}^{n}E and ˆ⊗_{μ, s}^{n}E.

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Holomorphic Functions on Domains in Operator Spaces

Lawrence Harris. University of Kentucky (USA)

Abstract. This talk will give an overview of facts and unsolved problems about bounded symmetric domains, Siegel domains of genus 2 and unbounded homogeneous domains in spaces of operators.

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Polynomial Topologies on a Banach Space

Jesús A. Jaramillo. Universidad Complutense de Madrid (Spain)

Abstract. Let X be a real Banach space. The weak polynomial topology w_{P} on X is defined as the weakest topology for which every polynomial on X is continuous. Then w_{P} is given by a family of translation-invariant semi-metrics, but in general it is not linear. Canonically related to w_{P}, we introduce the so called locally convex polynomial topology τ_{P} on X. We prove that τ_{P} is the finest locally convex topology which is coarser than w_{P}. Several properties of the seminorms defining τ_{P} are also studied. Finally, the convergence of sequences for w_{P} and τ_{P} is compared.

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KAM theory in Infinite Dimensional Spaces

Ricardo Pérez-Marco. UCLA (USA)

Abstract. KAM theory is one of the two major theories in Dynamical Systems developped during the XXth Century. Its scope reaches other fields as Riemannian Geometry, PDE and Functional Analisis. For example, it did motivate Hamilton Local Inversion Theorem in Good Frechet spaces. Our purpose is to describe recent developments of KAM theory to Hamiltonian systems in Infinite Dimensional spaces.

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Polynomials on Banach Spaces with Unconditional Schauder Bases

Raymond A. Ryan. National University of Ireland (Ireland)

Abstract. Let X be a Banach space with an unconditional Schauder basis. We consider the problem of representing a homogeneous polynomial on a X by an unconditionally convergent monomial expansion. A recent result of Defant and Kalton shows that, in infinite dimensions, the space of n-homogeneous polynomials does not have an unconditional basis. However, it may be possible to represent a homogeneous polynomial by a monomial expansion that is pointwise unconditionally convergent. For example, when X = ℓ_{1}, every homogeneous polynomial has this property. Matos isolated the spaces of n-homogeneous polynomials that have pointwise unconditionally convergent monomial expansions and defined a complete norm for these spaces. We explore some properties of the Matos spaces, including a connection with absolutely summing operators.

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Integral Representation of Holomorphic Functions on Banach Spaces

Ignacio Zalduendo. Universidad Torcuato di Tella (Argentina)

Abstract. The Cauchy integral formula has no true analogue in infinite-dimensional holomorphy. The usual generalisation, though quite useful, is essentially the onedimensional formula in each direction: for each x, and |z| < r,

f(zx) = \frac{1}{2πi} ∫_{|λ|=r} \frac{f(λx)}{λ − z}dλ.

However, integral expressions valid for some homogeneous polynomials and some holomorphic functions have been proposed. All of them involve integration over E′ rather than E. An integral k-homogeneous polynomial over E is

P (x) = ∫_{B_{E′}} γ(x)^{k}dμ(γ),

and an integral holomorphic function f : B_{E}^{∘} ⟶ C is

f(x) = ∫_{B_{E′}} \frac{1}{1 − γ(x)}dμ(γ).

In these expressions, the measure μ is said to represent the polynomial P or the function f, but there are many such representing μ’s, and little has been said in the way of expressing these measures in terms of the functions and a universal measure G, i.e.: dμ(γ) = f̃(γ)dG(γ).

In this talk we discuss this problem for a class of entire functions of exponential type over a Banach space E with separable dual. We obtain integral representations

FUNCTION THEORY ON INFINTE DIMENSIONAL SPACES VIII 7

of the form

f(x) = ∫_{E′} e^{γ(x)}f̃(γ)dG(γ)

where G is a Gaussian measure on E′ and f̃ is a transformation of f involving the covariance operator of G.

SHORT TALKS

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On the Müntz Theorem for numerable compact sets

José María Almira. Universidad de Jaén (Spain)

Abstract. In this note we solve the Müntz problem for the space C(K) whenever K ⊂ [0, ∞) is a numerable compact set which satisfies certain additional assumptions and we propose the general case as an open question. By using a well known consequence of the Hahn-Banach theorem, we convert the problem of the density of the span of the monomials {x^{λ_{i}}}_{i=0}^{∞} in a problem about the zero sets of certain analytic functions. The open problem can thus be formulated as a basic question in the theory of complex functions.

References

[1] J. A. Clarkson and P. Erdös, Approximation by polynomials, Duke Math. J. 10 (1943) 5-11.

[2] L. Schwartz, Etude des sommes de’Exponentielles, Herman (Paris) (1959)

[3] P. Borwein, T. Erdelyi, The full Müntz theorem in C[0, 1] and L_{1}[0, 1], J. London Math. Soc. 54 (2) (1996) 102-110.

[4] P. Borwein, T. Erdélyi, Polynomials and polynomial inequalities, Graduate Texts in Mathematics, Springer (1996).

[5] A. Pinkus, Weierstrass and Approximation Theory, J. Approx. Theory 107 (2000), 1-66.

[6] O. Szàsz, Über die approximation stetiger funktionen durch lineare aggregate von potezen, Math. Ann. 77 (1916), 482-496.

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Radon-Nikodým compact spaces and Banach spaces of low weight

Antonio Avilés. Universidad de Murcia (Spain)

Abstract. The concept of Radon-Nikodým compact space has its origin in Banach space theory, and it is defined as a topological space which is homeomorphic to a weak^{∗} compact subset of the dual of an Asplund space (that is, a dual Banach space with the Radon-Nikodým property). It is an open problem whether a continuous image of a Radon-Nikodým compact is a Radon-Nikodým compact. We present a new partial answer to this problem: continuous images of

Radon-Nikodým compacta of weight less than cardinal b are Radon-Nikodým. Cardinal b is the least cardinal of a subset ω^{ω} not contained in a σ-compact

subset. It is consistent that b > ω_{1} and the statement b = c is weaker than Martin’s axiom.

This result on continuous images has a Banach space counterpart: If a Banach space of density character less than b is a subspace of an Asplund generated space (respectively a subspace of a WCG space), then it is Asplund generated (respectively WCG). There are counterexamples to this statements of density character exactly b.

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Foundations of complex Analysis in non locally Convex Spaces

Aboubakr Bayoumi. King Saud University (Saudi Arabia)

Abstract. In fact, real development in Function Theory has been given after we have introduced several new concepts, especially the concept of Quasi-Differentiability which is stronger than Fréchet one. We have built up a complete theory for Function Theory in F-spaces which are not necessarily locally convex.

This lecture will be on a part of our first book entitled: “Foundations of complex Analysis in non locally Convex Spaces” which appear this year.

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Operators with common Hypercyclic subspaces

Juan Bes. Bowling Green University (USA)

(Joint with R. Aron, F. Leon and A. Peris)

Abstract. Let T be a bounded operator on a Banach space X. A vector x in X is said to be hypercyclic for T provided the orbit {x, T x, T^{2}x, … } is dense in X. We consider conditions for a family of operators acting on X to have a common hypercyclic subspace, that is, to have a closed subspace of infinite dimension consisting entirely, except for the origin, of vectors that are hypercyclic for each operator in the family. Our work complements recent results by G. Costakis and M. Sambarino and by F. Bayart.

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Automatic continuity of directional derivatives

Eusebio Corbacho. Universidad de Vigo (Spain)

Joint work with V. Tarieladze

Abstract. Let f be a mapping defined on an open set G of a Banach space E with values in a Banach space F and x_{0} ∈ G. Definition. The mapping f is said to be directionally differentiable x_{0} ∈ G if there is a mapping T_{x_{0},f} : E → F such that

(1)  T_{x_{0},f}(u) = lim_{t→0} \frac{f(x_{0} + th) − f(x_{0})}{t} ∀h ∈ E.

If f is directionally differentiable at x_{0} and its directional derivative T_{x_{0},f} : E → F is a continuous linear operator, then f is said to be Gâteaux differentiable at x_{0}.

We consider two natural questions: Question 1. Assume that the given continuous f is directionally differentiable at x_{0} and its directional derivative T_{x_{0},f} : E → F is an additive mapping. Is then f Gâteaux differentiable at x_{0}? Question 2. Is there a wide class Φ of mappings from G to F such that if a given continuous f ∈ Φ is the directionally differentiable at x_{0}, then f Gâteaux differentiable at x_{0}?

In the given talk we will discuss the following two results: Theorem 1. Question 1 has the affirmative answer. Theorem 2. If G is convex, F = R and Φ consists of convex functionals, then Question 2 has the affirmative answer as well.

Theorem 1 seems not to be noticed earlier in the literature. Theorem 2 is known [G, p. 127, Theorem 7 ].

This talk is based on [BCPT], where more general versions of Theorem 1 and Theorem 2 are contained.

REFERENCES

[BCPT] Banakh, T., Corbacho, E., Plichko, A., Tarieladze V., Automatic continuity and linearity of Gâteaux derivatives, In preparation.

[G] Giles, J.R. Convex analysis with applications in differentation of convex functions. Pitman, London, 1982, 278 p.

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Approximation of uniformly continuous mappings in Orlicz spaces

Sylvain Delpech. Université de Bordeaux (France)

Abstract. We estimate the best exponent α for which every uniformly continuous function, from the unit ball, B_{M}, of a reflexive Orlicz functions space, endowed with the Luxemburg norm ∥ . ∥_{M}, into any normed vector space (Y, ∥ . ∥_{Y}), can be approximated by α-Hölder functions. We also discuss the sharpness of our estimate. Then we prove that this best α, defined by :

α(B_{M}, Y) = sup\{α ∈ [0, 1], \overline{H^{α}(B_{M}, Y)}^{∥ . ∥_{∞}} = UC(B_{M}, Y)\},

where H^{α}(B_{M}, Y) is the set of α-Hölder functions between B_{M} and Y, UC(B_{M}, Y) is the set of uniformly continuous functions between B_{M} and Y, may sometimes be only a supremum and not a maximum.

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Viscosity solutions to Hamilton-Jacobi equations on Riemannian manifolds

Juan Ferrera. Universidad Complutense de Madrid (Spain)

Joint work with D. Azagra and F. López Mesas

Abstract. We establish some perturbed minimization principles, and we develop a theory of subdifferential calculus, for functions defined on Riemannian manifolds. Then we apply these results to show existence and uniqueness of viscosity solutions to Hamilton-Jacobi equations defined on Riemannian manifolds.

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Composition operators on a Hilbert space of Dirichlet Series

Catherine Finet. Université de Mons-Hainaut (Belgium)

Abstract. We study the numerical range and compacity of some composition operators, defined on a Hilbert space of Dirichlet series introduced by Hedenmalm, Linqvist and Seip. We describe the numerical range of “nice” composition operators. We focus on the zero-inclusion question. The compacity is studied for symbols

ϕ(s) = c_{0}s + c_{1} + ∑_{j=1}^{d} c_{q_{j}}q_{j}^{−s} (d = 1, 2, 3)

where the q_{j}’s are multiplicatively independent. The situation is completely different, according whether c_{0} ≥ 1 or c_{0} = 0. This is a joint work with H. Quefflec and A. Volberg.

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Smooth Approximation on Banach Spaces

Robb Fry. Saint Francis Xavier University (Canada)

Abstract. We consider the general problem of uniformly approximating continuous functions on Banach spaces by functions with a higher degree of smoothness.

We shall discuss some of the key classical results, as well as more recent and less known developments.

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Polynomials generated by linear operators

Pablo Galindo. Universidad de Valencia (Spain)

Joint work with M. L. Lourenco and L. A. Moraes

Abstract. The class of Banach algebra valued n-homogeneous polynomials generated by the n^{th} powers of linear operators is the topic of this talk. We compare it with the finite type polynomials, as well as their closures in the norm topology.

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Optimal Lipschitz Extensions

Thierry Gaspari. Université de Bordeaux (France)

Abstract. Let X be a Banach space, Ω be an open subset of X and u ∈ C(Ω, ℝ). We say that u is AML (Absolutely Minimizing Lipschitzian) if for all bounded open set V with \overline{V} ⊂ Ω, for all v ∈ C(\overline{V} , ℝ), we have

u = v on ∂V ⇒ Lip(u, V) ≤ Lip(v, V).

If u is AML in Ω and equals the function f on the boundary of Ω, then u is called an optimal Lipschitz extension of f in Ω. We study the relationship between the AML functions, the viscosity solutions of the infinity Laplacian partial differential equation, and the functions which satisfy another property called the comparison with cones. We prove that these notions are equivalent in any Hilbert space X. We also study the existence of optimal Lipschitz extensions.

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Peripheral spectrum of positive operators and generalizations of Jentzsch’s theorem

Arkady Kitover. Community College of Philadelphia (USA)

Abstract. The main result. Theorem. Let X be a Banach lattice and T : X → X be a positive operator. If the peripheral spectrum of T consists of a finite number of eigenvalues of T then it is cyclic.

Moreover, if X has the Fatou property, T is sequetially order continuous, and for any non-zero positive u ∈ X T u is a weak unit in X then the peripheral spectrum consists of only one point r(T) and the corresponding eigenspace is one-dimensional.

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Average locally uniformly rotund dual norms

Sebastián Lajara. Universidad de Castilla La Mancha (Spain)

Abstract. Average locally uniformly rotund norms (ALUR for short) are a generalization of LUR norms. They were introduced by Troyanski, who proved that every rotund norm with the Kadec property is ALUR and that every ALUR space admits an equivalent LUR norm. This result suggests the following problem: If a dual Banach space is ALUR, does it necessarily admit an equivalent dual LUR norm? The dual of the James Tree space provides a negative answer to this question (Schachermayer). Therefore, the class of dual LUR spaces is strictly contained in the class of ALUR dual renormable spaces. Banach spaces with equivalent LUR norms were characterized in terms of

countable decompositions of such spaces by Moltó, Orihuela

and Troyanski, and by Raja, who gave a generalization to the dual case. In this note, we provide a covering type characterization of dual Banach spaces with equivalent ALUR dual norms.

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Canonical isometries between spaces of polynomials

Silvia Lassalle. Universidad de Buenos Aires (Argentina)

Abstract. This talk is concerned with work with Christopher Boyd in which isometries between spaces of homogeneous polynomials are studied. We characterize the case of P_{A}(^{n}E) when E is a real Banach space. In the complex case a better knowledge of the extreme points of the unit ball of P_{I}(^{n}E) is required. The characterization of the isometries of P_{A}(^{n}E) when E is a real Banach space can be extended to give a characterization of the isometries of P_{A}(^{n}E, X) where X is a Banach space with X′ strictly convex. This result replies heavily on the idea of centralizer of a Banach space.

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Spectra of weighted composition operators on weighted

Banach spaces of analytic functions

Joint work with Richard Aron

Mikael Lindström. Åbo Akademi University (Finland)

Abstract. We determine the spectra of weighted composition operators acting on the weighted Banach spaces of analytic functions H_{v_{p}}^{∞} when the symbol ϕ has a fixed point in the open unit ball. As an application of this result we give the spectra of composition operators on Bloch type spaces.

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Linear polarization constant of ℝ^{n}

Mate Matolcsi. Renyi Institute for Mathematics (Hungary)

Abstract. In this note we aim to make a contribution to estimating the n-th linear polarization constant c_{n}(H) of an n-dimensional real Hilbert space H. We recall the

Definition 1 (Benítez, Sarantopoulos, Tonge). The n-th linear polarization constant of a Banach space X is defined by

c_{n}(X) := inf{M : ∥f_{1}∥ ⋯ ∥f_{n}∥ ≤ M∥f_{1} ⋯ f_{n}∥ (∀f_{1}, …, f_{n} ∈ X^{∗})}
= 1/ inf_{f_{1},...,f_{n}∈S_{X^{∗}}} sup_{∥x∥=1} |f_{1}(x) ⋯ f_{n}(x)|.

For a real or complex Hilbert space H of dimension at least n, we always have c_{n}(H) = c_{n}(𝕂^{n}).

Bení tez, Sarantopoulos and Tonge proved that for isomorphic Banach spaces X and Y we have c_{n}(X) ≤ d^{n}(X, Y)c_{n}(Y), where d(X, Y) denotes the Banach-Mazur distance of X and Y. Note, that for any n-dimensional space X a result of John states that d(X, 𝕂^{n}) ≤ \sqrt{n} (where 𝕂^{n} denotes the n-dimensional Hilbert space). The combination of these results mean that the determination of c_{n}(𝕂^{n}) gives information on the linear polarization constants of other spaces, too.

The exact values of c_{m}(𝕂^{n}) seem very hard to determine. A remarkable result of Arias-de-Reyna states that c_{n}(ℂ^{n}) = n^{n/2}. Ball’s recent solution of the complex plank problem also implies the same result.

The value of c_{n}(ℝ^{n}) seems even harder to find. The determination of c_{n}(ℝ^{n}), by the definition and the Riesz representation theorem, boils down to determining

I := inf_{x_{1},...,x_{n}∈S} sup_{∥y∥=1} | ⟨x_{1}, y⟩ ⋯ ⟨x_{n}, y⟩ |

The estimate I ≤ n^{− \frac{n}{2}} follows by considering an orthonormal system.

The result of Arias-de-Reyna can be used to derive the following estimates:

n^{\frac{n}{2}} ≤ c_{n}(ℝ^{n}) ≤ 2^{\frac{n}{2} −1}n^{\frac{n}{2}} .

A natural, intriguing conjecture is the following.

Conjecture. c_{n}(ℝ^{n}) = n^{n/2}.

Marcus gives the following estimate: If x_{1}, x_{2}, …, x_{n} are unit vectors in ℝ^{n} then there exists a unit vector y such that

(2)  | ⟨x_{1}, y⟩ ⋯ ⟨x_{n}, y⟩ | ≥ (λ_{1}/n)^{n/2},

where λ_{1} denotes the smallest eigenvalue of the Gram matrix XX^{∗} = [⟨x_{i}, x_{j}⟩]. Marcus also expressed the opinion that lower bounds on sup_{∥y∥=1} | ⟨x_{1}, y⟩ ⋯ ⟨x_{n}, y⟩ | should involve the eigenvalues λ_{1}, …, λ_{n} of the Gram matrix XX^{∗}, i.e. we should look for estimates of the form sup_{∥y∥=1} | ⟨x_{1}, y⟩ ⋯ ⟨x_{n}, y⟩ | ≥ f(λ_{1}, …, λ_{n})n^{−n/2}.

Note that ∑_{j} λ_{j} = Tr XX^{∗} = n. Therefore the above Conjecture can be equivalently formulated as

sup_{∥y∥=1} | ⟨x_{1}, y⟩ ⋯ ⟨x_{n}, y⟩ | ≥ 1 · n^{−n/2} = (\frac{λ_{1} + ⋯ + λ_{n}}{n} )^{n/2} n^{−n/2}.

In this note we show that if we replace the arithmetic mean by the harmonic mean of the numbers λ_{1}, …, λ_{n}, then the corresponding lower estimate does hold.

Theorem. 1. Assume that the given unit vectors x_{1}, …, x_{n} are linearly independent. Let λ_{1}, …, λ_{n} denote the eigenvalues of the Gram matrix XX^{∗}. Then

(3)  max_{∥y∥=1} ∏_{j=1}^{n} |(Xy)_{j}| ≥ (\frac{n}{λ_{1}^{−1} + ⋯ + λ_{n}^{−1}} )^{n/2} · n^{n/2}

This gives an improvement of Marcus’ result but the Conjecture still remains open.

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Finite-Dimensional Banach Spaces with numerical index zero

Javier Merì. Universidad de Granada (Spain)

Abstract. We prove that a finite-dimensional Banach space X has numerical index 0 if and only if it is the direct sum of a real space X_{0} and nonzero complex spaces X_{1}, …, X_{n} in such a way that the equality

∥x_{0} + e^{iq_{1}ρ}x_{1} + ⋯ + e^{iq_{n}ρ}x_{n}∥ = ∥x_{0} + ⋯ + x_{n}∥

holds for suitable positive integers q_{1}, …, q_{n}, and every ρ ∈ ℝ and every x_{j} ∈ X_{j} (j = 0, 1, …, n). If the dimension of X is two, then the above result gives X = ℂ, whereas dim(X) = 3 implies that X is an absolute sum of ℝ and ℂ. We also give an example showing that, in general, the number of complex spaces cannot be reduced to one.

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El operador de Volterra no es superciclico.

Alfonso Montes. Universidad de Sevilla (Spain)

Abstract. En esta charla veremos un esbozo de la prueba de que el operador de clasico de Volterra no es superciclico. En la prueba se hace un uso especial de los polinomios de Legendre. Trabajo conjunto con Eva Gallardo.

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Deleting diffeomorphisms and starlike bodies in Banach spaces

Alejandro Montesinos. Universidad Complutense de Madrid (Spain)

Abstract. This talk aims to show the theory of topological negligibility in its historical context, and to explore, with some detail, both the way in which it has developed and its interesting applications in several branches of mathematics. Given a topological space X and a subset K of X, we say that K is topologically negligible if there exists a homeomorphism between X and X \K. This homeomorphism, which we call deleting, is usually required to be the identity outside a given neighborhood of K. In 1953, V. Klee proved that if X is a non-reflexive Banach space or an infinite-dimensional L^{p} space, every K compact subset of X is topologically negligible. Later on, several mathematicians (C. Bessaga, T. Dobrowolski and others) studied the smooth negligibility, where, by discarding deleting homeomorphims, the attention is paid to the notion of deleting diffeomorphims. For the time being, the deepest results about smooth negligiblity are due to D. Azagra and T. Doborwolski, who showed that for every Banach space (X, ∥ · ∥) with a C^{p}-smooth norm ρ, compacta of X are negligible via a C^{p}-diffeomorphism. Now, in a joint work with D. Azagra, we have proved, by exploiting starlike bodies, some other results that contribute to understand in what Banach spaces, compacta and points are smooth negligible. For instance, we have proved that in every Banach space X with smooth partitions of unity, weak compacta of X are smooth negligible. In addition, it has been established that if X is an infinite-dimensional separable Banach space with a Schauder basis, and X has a C^{p}- smooth bump function, then for every compact subset K and every open subset U of X with K ⊂ U, there exists a C^{p}-diffeomorphism h : X → X \ K such that h is the identity on X \ U.

Finally, it is important to remark that this kind of results about topological and smooth negligibility has found interesting applications in several branches of mathematics, which include fixed point theory, strange phenomena concerning EDO’s, and approximate strong versions of the Morse-Sard theorem.

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Compacta in product of lines from a functional-analytic point of view

Vicente Montesinos. Universidad de Valencia (Spain)

Joint work with M. Fabian, G. Godefroy and V. Zizler

Abstract. Along several papers we developed a technique of splitting certain subsets of a Banach space X in such a way that it allows us to detect whether or not X belongs to one of several classes of non-separable Banach spaces. Those subsets sometimes are Markushevich bases, sometime just total and countably supported subsets. Classes of Banach spaces considered included the weakly compactly generated, the subspaces of weakly compactly generated, the Vašák or weakly countably determined and the weakly Lindelöf determined. The technique is reminiscent of the way several authors (S. Argyros, S. Mercourakis, V. Farmaki, S. P. Gu’lko, G. A. Sokolov,...) used to split the set of coordinates in representing compact sets as subsets of products of lines. It is not surprising, then, that we can recover those representations by using our approach. This is done here for the Eberlein and the Gul’ko compacta.

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Vector valued Sobczyk’s theorem

Yolanda Moreno. Universidad de Extremadura (Spain)

Abstract. We provide a vectorial version of Sobczyk’s theorem. Recall that Sobczyk’s theorem can be read as follows: every w^{∗}-null sequence of functionals (y_{n}^{∗})_{n} on a Banach space, say Y, can be extended to any space X such that X/Y is separable by a w^{∗}-null sequence of functionals. We prove a vectorial version of such result, essentially: every uniformly bounded sequence of operators T_{n} : Y → Y_{n} admitting uniformly bounded extensions to a space X such that X/Y is separable and in such a way that (T_{n}) converges to 0 in the strong operator topology (SOT) can be extended to X by a SOT -null sequence. In particular, Sobczyk’s theorem and the Rosenthal and Johnson-Oikhberg’s results related to the topic are obtained as consequence.

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Kuratowski’s index of non-compactness and renorming in Banach spaces

Luis Oncina. Universidad de Murcia (Spain)

Joint work with F. García, J. Orihuela and S. Troyanski

Abstract. A point x ∈ A ⊂ (X, ∥ · ∥) is quasi-denting if for every ε > 0 there exists a slice of A containing x with Kuratowski index less than ε, where the Kuratowski index of non-compactness of B is defined by α(B) := inf{ε > 0 : B is covered by a finite family of sets of diameter less than ε}.

Theorem. Let X be a normed space and let F be a norming subspace of its dual. Then X admits an equivalent σ(X, F)-lower semi-continuous LUR norm if, and only if, for every ε > 0 we can write X = ∪_{n=1}^{∞}X_{n,ε} in such a way that for every x ∈ X_{n,ε}, there exists a σ(X, F)-open half space H containing x with α (H ∩ X_{n,ε}) < ε.

Taking advantage of homogeneity one can replace the space X in the former theorem by its unit sphere.

¿From the topological point of view we shall see that a normed space X admits an equivalent σ(X, F)-lower semi-continuous LUR norm if, and only if, the norm topology has a network N = ∪_{n=1}^{∞}N_{n} such that for every n ∈ ℕ and for every x ∈ ∪N_{n} there is a σ(X, F)-open half space H with x ∈ H such that H meets only a finite number of elements from N_{n}, therefore turning the ‘discrete’ condition appearing in [A. Moltó, J. Orihuela, S. Troyanski, and M. Valdivia, On weakly locally uniformly rotund Banach spaces, J. Funct. Anal. 163 (1999), no. 2, 252– 271.] into a ‘locally finite’ one.

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Spectra of composition operators on BMOA

Niklas Palmberg. Åbo Akademi University (Finland)

Joint work with Mikael Lindström (Åbo Akademi University)

Abstract. It is shown that if ϕ is an univalent self map on the unit disk D, not an automorphism, with a fixed point in D and the essential spectral radius of the composition operator C_{ϕ} on H^{2} is different from zero, then the spectrum of C_{ϕ} on BMOA coincides with \overline{D}. This answers in the affirmative a conjecture by MacCluer and Saxe.

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Absolutely Summing Multilinear Operators

David Pérez. Universidad Rey Juan Carlos (Spain)

Abstract. We give a new definition of absolutely summing multilinear operator and we see that it seems to be the right one, in the sense that we can generalize most of the linear results. In particular, we have multilinear versions of Grothendieck’s Theorem and good relations with some other well known classes of multilinear operators, such as Hilbert-Schmidt, nuclear or integral operators.

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Hamiltonian structure of the harmonic functions on an annulus

Luis M. Pozo Coronado. Universidad Complutense de Madrid (Spain)

Joint work with J. Muñoz Masqué

Abstract. We consider the space of harmonic functions on an n-dimensional annulus, seen as the space of extremals of the energy functional. The variational approach allows us to find an alternate bilinear form which is proved to be nondegenerate, thus endowing the harmonic functions with an structure of a symplectic manifold.

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On lineability of sets of unimaximal continuous functions

Lucas Quarta. Université de Mons-Hainaut (Belgium)

Abstract. A set M in a linear topological space X is said to be n-lineable (resp. lineable, resp. spaceable) in X if M ∪ {0} contains a vector space of dimension n (resp. infinite dimensional, resp. closed and infinite dimensional). If the maximum cardinality of such a vector space exists it is called the lineability of M. One can find in [1] a lot of results of “linearity in non linear problems” in many different fields of analysis. One of the first works in this spirit is the lineability of the set of nowhere differentiable functions on [0, 1],[6] (see also [4], [7] and [8]). Recently, several papers were devoted to the study of the lineability of sets of functions on [0, 1] or ℝ which satisfy other special properties (see for example [2] and [3]). Our goal is the study of the lineability of the set M = Ĉ(S) of continuous functions defined on a set S and which admit one and only one absolute maximum. We will present an analytical approach when S is a subset of ℝ (joint work with V.I. Gurariy) and a geometrical one (based on ideas introduced in [5]) in higher dimensions (joint work with C. Troestler).

References

[1] R. Aron, D. Garcia, M. Maestre, Linearity in non linear problems, Rev. R. Acad. Cien. Serie A. Mat. Vol. 95(1), 2001, 7–12.

[2] R. Aron, V. I. Gurariy, J. Seoane, Lineability and spaceability of sets of functions on ℝ, to appear in Proc. Amer. Math. Soc.

[3] P. Enflo, V. I. Gurariy, On lineability and spaceability of sets in function spaces, preprint.

[4] V. Fonf, V. I. Gurariy, V. Kadec, An infinite dimensional subspace of C[0,1] consisting of nowhere differentiable functions, C. R. Acad. Bulgare Sci. 52 (1999), No 11–12, 13–16.

[5] G. Godefroy, J. Saint Raymond, Sous-espaces de dimension 3 des espaces C(K), Bull. Sc. math., 2ème série, 103 (1979), 259–262.

[6] V. I. Gurariy, subspaces and bases in spaces of continuous functions, (Russian) Dokl. Akad. Nauk SSSR 167 (1966), 971–973.

[7] V. I. Gurariy, Linear spaces composed of nondifferentiable functions, C. R. Acad. Bulgare Sci. 44 (1991), No 5, 13–16.

[8] L. Rodriguez-Piazza, Every separable Banach space is isometric to a space of continuous nowhere differentiable functions, Proc. Amer. Math. Soc. 123, 12 (1995), 3649–3654.

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Polarization constants, plank problems, and local theory of Banach spaces

Szilard Revesz. Renyi Institute for Mathematics (Hungary)

Joint work with Y. Sarantopoulos

  Abstract.

Definition 2 (Benítez - Sarantopulos - Tonge). The n^{th} (linear) polarization constant of a Banach space X is defined by

(4) c_{n}(X) := inf{M > 0 : ∥f_{1}∥ ⋯ ∥f_{n}∥ ≤ M · ∥f_{1} ⋯ f_{n}∥, ∀f_{1}, …, f_{n} ∈ X^{∗}} .

Moreover, the (linear) polarization constant of X is c(X) := lim sup_{n→∞} c_{n}(X)^{1/n}.

If f_{1}, f_{2}, …, f_{n} ∈ X^{∗}, then the product (f_{1}f_{2} ⋯ f_{n})(x) := f_{1}(x)f_{2}(x) ⋯ f_{n}(x) is a continuous n-homogeneous polynomial on X. In 1998 R.A. Ryan and B. Turett proved the existence of a universal bound C_{n} of (4), and Benítez, Sarantopoulos and Tonge proved that in the case of complex Banach spaces, C_{n} ≤ n^{n} and the constant n^{n} is best possible. Clearly, c_{n}(·) and c(·) are isometric properties of Banach spaces, also fitting into the local theory of Banach spaces, since for any X we have c_{n}(X) = sup {c_{n}(Y) : Y ⊆ X, dim Y = n}.

Proposition 1. For c(X) also the limit exists and we have

c(X) = lim_{n→∞} c_{n}(X)^{1/n} .

The growth of c_{n}(X) when n → ∞ is characteristic to the space X. Eg. we have

Theorem. 2. Let X be any normed space. Then c(X) = ∞ iff dim(X) = ∞.

Although it seems to be very likely, but it is not known whether c_{n}(X) ≥ c_{n}(ℓ_{2}^{n}) (∀n ≤ dim X) holds true in general. Still, for each n ∈ ℕ, Hilbert spaces have the smallest n^{th} polarization constant among infinite dimensional normed spaces.

Theorem. 3. If dim X = ∞, then c_{n}(X) ≥ c_{n}(ℓ_{2}^{n}) , ∀n ∈ ℕ.

In this work we discuss various results, estimates and exact values found for linear polarization constants of Banach, and in particular of Hilbert spaces. We also present expected and unexpected connections to the plank problem of Tarski, to Chebyshev constants, and to various well-known polynomial inequalities. In this respect, one of the challanging problems of the subject is the following conjecture, which seems to be widely believed.

Conjecture [Sarantopoulos]. c_{n}(ℝ^{n}) = n^{\frac{n}{2}} .

Note that Arias-de Reyna proved c_{n}(ℂ^{n}) = n^{\frac{n}{2}} , and an even more precise result was found by K. Ball recently. We will explain, why the real case is more difficult.

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The Birkhoff integral

José Rodríguez. Universitad de Murcia (Spain)

Abstract. There are several extensions of Lebesgue’s theory of integration to the case of functions with values in Banach spaces, being widely known those due to Bochner and Pettis. Birkhoff integral lies strictly between the Bochner and Pettis integrals and has caught the attention of some authors pretty recently. This talk will be devoted to show the new results about this topic that appear in our papers: “Birkhoff integral and the property of Bourgain” (with B. Cascales) and “On the existence of Pettis integrable functions which are not Birkhoff integrable”.

We prove that there is a close relationship between Birkhoff integrability and Bourgain property: a bounded function f defined on a complete probability space with values in a real Banach space X is Birkhoff integrable if, and only if, the set of compositions of f with elements of the dual unit ball has Bourgain property. This fact allows us to characterize the weak Radon-Nikodým property in dual Banach spaces via Birkhoff integrable Radon-Nikodým derivatives. As a consequence we give a solution, in the case of dual Banach spaces, to a problem posed by Fremlin concerning the representation of vector measures as indefinite Pettis integrals of generalized McShane integrable functions.

It is well known that when the range Banach space is separable, then Birkhoff and Pettis integrability coincide. We try to go a bit further when studying the differences between Birkhoff and Pettis integrability in non-separable Banach spaces. We prove that for a weakly Lindelöf determined Banach space X with density character greater than or equal to the continuum, there exists a bounded Pettis integrable function f : [0, 1] ⟶ X which is not Birkhoff integrable. A particular case of our constructions shows that the analogue of Lebesgue’s dominated convergence theorem for the Birkhoff integral does not hold in general.

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Strictly singular inclusions between symmetric sequence spaces

Víctor M. Sánchez. Universidad Complutense de Madrid (Spain)

Joint work with F.L. Hernández and E.M. Semenov

Abstract. Strict singularity of inclusions between symmetric sequence spaces will be studied. Suitable conditions will be provided involving the associated fundamental functions. The special case of Lorentz and Marcinkiewicz spaces will be characterized. It will be also showed that if E ↪ F are symmetric sequence spaces with E ≠ ℓ_{1} and F ≠ c_{0} and ℓ_{∞} then there exist a intermediate symmetric sequence space G such that E ↪ G ↪ F and both inclusions are not strictly singular. As a consequence new characterizations of the spaces c_{0} and ℓ_{1} inside the class of all symmetric sequence spaces will be given.

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Banach Spaces and Polytopes

Antonio Santos. Universidad de Vigo (Spain)

Abstract. Let X be a Banach Space and B(X) the unit ball of X. If B(X) is a polyhedron then B(X^{∗}) is a polyhedron? Where B(X^{∗}) is a unit ball of the dual space.

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Construction of Infinite Dimensional Vector Spaces of Hypercyclic Vectors

Juan B. Seoane. Kent State university (USA)

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

Abstract. Consider X = ℓ_{p} or c_{0}, and define the weighted backward shift B_{λ} (|λ| > 1) on X,

B_{λ}(x_{1}, x_{2}, x_{3}, …) = λ · (x_{2}, x_{3}, x_{4}, …).

We give a different proof that the set of hypercyclic vectors for this operator is lineable, i.e. there exists an infinite dimensional hypercyclic subspace for B_{λ}. We also give a technique to construct infinite dimensional common hypercyclic subspaces for a countable family of these operators. This is a well known result, but we won’t use the Baire category theorem, we prove it in an explicit and constructive way.

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Strong minimality of Gaussian integrals

Ricardo Vidal Vázquez. Universidad de Vigo (Spain)

Joint work with V. Tarieladze

Abstract. Let E be a be a finite-dimensional Hilbert space with dim(E) ≥ 1, U(E) be the set of all isometric linear operators u : E → E and γ be the standard Gaussian measure on E.

We will discuss some applications of the next statement which expresses “the strong minimality” of the Gaussian integrals. Proposition. Let μ be a positive measure (not necessarily finite) given on the Borel σ-algebra of E such that

∫_{E} ∥x∥^{2}dμ(x) = 1.

Then for any positively 2-homogeneous Borel measurable function f : E → C with f(0) = 0, we have:

∫_{E} |f(x)|dγ(x) ≤ dim(E) sup_{u∈U(E)} ∫_{E} |f(ux)|dμ(x)

holds. This talk is based on [TV].

REFERENCES

[TV] Tarieladze V., Vidal, R. Strong minimality of Gaussian summing norm, Acta Mathematica Vietnamica, Vol. 28, Number 2, 2003, pp. 225-240.

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Where do homogeneous polynomials on ℓ_{1}^{n} attain their norm?

Ignacio Villanueva. Universidad Complutense de Madrid (Spain)

Joint work with David Pérez (Universidad Rey Juan Carlos)

Abstract. In the “Open Problems” session of the conference ‘Function Theory on Infinite Dimensional Spaces VII’, held in Madrid in 2001, Professor I. Zalduendo asked the question of ‘how many’ homogeneous polynomials will attain their norm at the vertices of the unit ball of ℓ_{∞}^{n} when n tends to infinity. He conjectured that ‘almost everyone’. In this direction, he and D. Carando published recently a paper giving qualitative general results. As they say in the introduction, the question is to study how likely it is for a polynomial P : E ⟶ ℝ to attain its norm at a given subset A of the unit ball B_{E}.

In our talk we present quantitative results referring to 2-homogeneous poynomials on ℓ_{1}^{n}, as an example of the results that can be expected in more general cases: Using a ‘reasonable’ measure in P(^{2}ℓ_{1}^{n}), the space of 2-homogeneous polynomials on ℓ_{1}^{n}, we show the existence of a set of positive (and independent of n) measure of polynomials which do not attain their norm at the vertices of the unit ball of ℓ_{1}^{n}. Next we prove that, when n grows, almost every polynomial attains its norm in a face of ‘low’ dimension.

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