BOOK OF ABSTRACTS
FUNCTION THEORY ON INFINITE DIMENSIONAL SPACES
X
Madrid, 11-14 December 2007 Facultad de Ciencias Matem´ aticas Departamento de An´ alisis Matem´ atico
Universidad Complutense de Madrid SPAIN
Organizing Committee:
Jes´us ´Angel Jaramillo Aguado Gustavo Adolfo Mu˜noz Fern´andez Angelines Prieto Yerro
Juan Benigno Seoane Sep´ulveda
Index of Abstracts
(In this index, in case of multiple authors only the speaker is shown)
Plenary Talks
Juan Carlos ´Alvarez Paiva Dual normed spaces have the same girth 4 Luigi Ambrosio Existence and stability results of Fokker-Planck
equations and Markov processes associated to log- concave measures
4
Richard M. Aron Norm divergent, weakly dense sequences 5 Robert Deville Almost classical solutions of Hamilton Jacobi
equations
5
Albert Fathi An Introduction to Weak KAM Theory 6
Pablo Galindo Interpolating sequences in spaces of bounded ana- lytic functions
6 Domingo Garc´ıa The Bishop-Phelps-Bollob´as theorem for opera-
tors.
6 Gilles Godefroy Isometrically universal Banach spaces 7 Alberto Ibort On the extensions of a class of generalized K-
symmetric operators
7 Jos´e Orihuela James boundaries, selectors, and risk measures
applications
8 Alfred Peris Hypercyclic operators: When the linear dynamics
becomes chaotic
8 Raymond Ryan Regular Holomorphic Functions on Complex Ba-
nach Lattices
9 Juan B. Seoane Sep´ulveda Geometry of Polynomials in Banach spaces and its
applications to Bernstein and Markov inequalities 9
Ignacio Zalduendo Linearization and Compactness 10
Short Talks
J. Araujo Stability and instability of operators be- tween spaces of continuous functions
11 J. Cabello S´anchez Multiplicative bijections of Lipschitz and
smooth functions
11 D. Carando Spectra of weighted algebras of holomorphic
functions
12 J. A. Conejero Hypercyclic and chaotic behaviour of
C0-semigroups generated by Ornstein- Uhlenbeck operators
12
E. Durand Cartagena A Banach-Stone theorem for D∗ spaces 13 A. R. Dzhanoev Function of stabilization for dissipative dy-
namical systems
13 M. Fern´andez Unzueta Extension of polynomials as a local prop-
erty
14 I. Ferrando Tensor product representation of the
(pre)dual of order continuous p-convex Ba- nach lattices.
14
F. Gallego Lupia˜nez On M -spaces and Banach spaces 15 A. Gonz´alez Correa Flat sets, `p-generating and fixing c0 in
nonseparable setting
15 B. C. Grecu The relationship between extreme homoge-
neous polynomials and multilinear forms on Hilbert spaces
15
A. J. Guirao On the moduli of Squareness 16
O. Gut´u Global Inversion on Length Spaces 16
P. H´ajek Zero sets of real polynomials 16
B. Hernando Approximation numbers of nuclear and Hilbert-Schmidt multilinear forms defined on Hilbert spaces
17
M. Johanis Functions locally dependent on finitely many coordinates
17 E. Jord´a On bounded vector-valued holomorphic
funcions
17 S. Lajara The geometry of the bidual of a separable
Banach space
18
S. Lassalle Atomic decompositions for tensor products and polynomial spaces
18 P. Linares Orthogonally additive polynomials in `p 19 M. Lindstr¨om Essential norm of operators on weighted
Bergman spaces of infinite order
19 E. Mart´ın Peinador Fr´echet-Urysohn property for topological
groups
19 A. Miralles Hankel Operators on Algebras of Analytic
Functions
20 V. Montesinos Weak compactness and sigma-Asplund
generated spaces
20 G. A. Mu˜noz Fern´andez Unconditional constants in spaces of poly-
nomials
21
S. Muro Hypercyclic convolution operators on
Fr´echet spaces of analytic functions
21 D. P´erez Garc´ıa Bell inequalities: a long path from Einstein
to Pisier
22 D. Pinasco Lp representable functions on Banach
spaces
22 A. Proch´azka Winning tactics in a geometrical game 23 P. Rueda Biduality in weighted Banach spaces of
holomorphic functions
23 V. M. S´anchez de los Reyes Strictly singular inclusions into L1+ L∞ 24 Y. Sarantopoulos On the real Plank problem and its applica-
tions
24 S. M. Stoian A functional Calculus for Quotient
Bounded Operators
25
J. Su´arez Twisting Schatten classes 25
T. Ullrich Tensor Products of Besov Spaces and Ap- plications
25 A. Weber A class of hypercyclic Volterra composition
operators
26
PLENARY TALKS
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Dual normed spaces have the same girth
Juan Carlos ´Alvarez Paiva.
Universit´e des Sciences et Technologies de Lille, France.
Abstract. The girth of a normed space is defined as twice the length of the shortest curve on its unit sphere that joins a pair of antipodal points.
This interesting isometry invariant of normed paces was defined and studied by J.J. Schaeffer in the 70’s who left us with two interesting conjectures : (1) dual normed spaces have the same girth ; (2) if the girth of a normed space of dimension greater than 2 is greater or equal to 2π, then the space is Euclidean.
In this talk I will give an overview the proof of the first conjecture and some remarks on the second.
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Existence and stability results of Fokker-Planck equations and Markov processes associated to
log-concave measures
Luigi Ambrosio.
Scuola Normale Superiore. Pisa, Italy.
Abstract. In this talk, based on a joint work with L.Zambotti and G.Savare’, I will describe some applications of the variational theory of Fokker-Planck equations in the Wasserstein space of probability measures, first introduced by Jordan, Kinderlehrer and Otto. Developing some ideas contained in the book with Gigli and Savare’, we show that the Wasserstein formulation is sufficiently robust to allow for singular drift terms, and even infinite-dimensional state spaces. As a byproduct, general existence and sta- bility results can be obtained for the Fokker-Planck equations and for the associated Markov processes.
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Norm divergent, weakly dense sequences
Richard M. Aron.
Kent State University. USA.
Abstract. Essentially, we prove the result in the title. For any separa- ble, infinite dimensional Banach space E, there is a sequence (xn) in E with the following two properties:
1). kxnk → ∞, and
2). The weak closure of the set {xn | n ∈ N} is E.
The problem was motivated by a result of K. Chan and R. Sanders on weak, not norm, hypercyclic operators on `2. Joint with Domingo Garc´ıa and Manuel Maestre.
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Almost classical solutions of Hamilton Jacobi equations
Robert Deville.
Universit´e de Bourdeaux. France.
Abstract. We shall present a joint work with Jesus Jaramillo. Let Ω be a bounded open subset of Rd, F : Ω × Rd → R and u0 : ∂Ω → R be continuous. We say that a continuous function u : Ω → R is an almost classical solution of F (x, ∇u(x)) = 0 with Dirichlet condition u |∂Ω= u0 if :
u(x) = u0(x) for all x ∈ ∂Ω,
at each point x of Ω, u is differentiable and F (x, ∇u(x)) ≤ 0, and u satisfies F (x, ∇u(x)) = 0 for almost every x ∈ Ω.
While the existence of classical solutions imposes very severe conditions on F , we prove, under general hypotheses on F , the existence of almost classical solutions whenever d ≥ 2. In particular, there exists an almost classical solution of the Eikonal equation k∇u(x)k = 1 such that u(x) = 0 for all x ∈ ∂Ω if and only if d ≥ 2. Such solutions of the Eikonal equation cannot be C1-smooth, and are very different from the usual viscosity solution dist ( . , ∂Ω), which, of course, is not everywhere differentiable in Ω. We shall also present existence results of almost classical solutions of Hamilton-Jacobi equations defined on Riemannian manifolds.
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An Introduction to Weak KAM Theory
Albert Fathi.
Ecole Normale Sup´´ erieure de Lyon, France.
Abstract. Weak KAM Theory is about the deep connection between the Aubry-Mather Theory of Lagrangian Systems and the Theory of viscos- ity solutions of the Hamilton-Jacobi Equation. For the last ten years, it has been a fruitful line of thinking in several domains. We will give an elemen- tary introductory lecture on the subject explaining the connection that was found ten years ago.
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Interpolating sequences in spaces of bounded analytic functions
Pablo Galindo.
Universidad de Valencia, Spain.
Abstract. A sufficient condition for a sequence in the open unit ball of a complex Banach space E to be interpolating for H∞(BE) is presented: The sequence of their norms is an interpolating sequence for H∞. We will apply this result to the description of the spectra of some composition operators.
This a joint work with Alejandro Miralles.
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The Bishop-Phelps-Bollob´ as theorem for operators
Domingo Garc´ıa.
Universidad de Valencia, Spain.
Abstract. We prove the Bishop-Phelps-Bollob´as theorem for operators from an arbitrary Banach space X into a Banach space Y whenever the range space has property β of Lindenstrauss. We also characterize those Banach spaces Y for which the Bishop-Phelps-Bollob´as theorem holds for operators from `1 into Y . Several examples of classes of such spaces are provided. For instance, the Bishop-Phelps-Bollob´as theorem holds when the range space is finite dimensional, an L1(µ)-space for a σ-finite measure µ, a C(K)-space for a compact Hausdorff space K, or a uniformly convex Banach space. The material of this talk is based on a joint work with M. D.
Acosta, R. M. Aron and M. Maestre.
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Isometrically universal Banach spaces
Gilles Godefroy.
Universit´e Paris 6, France.
Abstract. In this joint work with Nigel Kalton, we show that a separa- ble Banach space which contains an isometric copy of every strictly convex separable Banach space, actually contains an isometric copy of every sepa- rable Banach space. The proof relies in part on descriptive set theory.
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On the extensions of a class of generalized K-symmetric operators
Alberto Ibort.
Universidad Carlos III, Spain.
Abstract. We introduce a class of operators on a Hilbert space H that contains both the well-known classes of symmetric, C-symmetric and formally-normal operators. Such operators will be called generalized K- symmetric operators and their definition is a natural extension of the pre- vious instances in the class of linear subpaces of the Hilbert space H2, thus extending Arens-Coddington theory. We will prove a general extension the- orem for such operators and we will characterize their maximal extensions.
As a result this theory generalizes both, von Neumann’s, Glazmann and Arens-Coddington theory. Moreover we will present sufficient conditions for the existence of maximal extensions of this class of operators that will contain as a particular instance a theorem by Galindo on the existence of a J -sefl-adjoint extension for a J -symmetric operator. Finally some topologi- cal properties of the space of extensions of generalized symmetric operators will be described.
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James boundaries, selectors, and risk measures applications
Jos´e Orihuela.
Universidad de Murcia, Spain.
Abstract. ABSTRACT NOT AVAILABLE.
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Hypercyclic operators: When the linear dynamics becomes chaotic
Alfred Peris.
Universidad Polit´ecnica de Valencia, Spain.
Abstract. During the last two decades the study of hypercyclic op- erators, on Banach or more general Fr´echet spaces, has developed into a very active research area. Hypercyclicity is one of the main ingredients in the most widely know definitions of chaos. Although chaos has long been thought of as being intrinsically linked to non-linearity, the investigations into hypercyclicity show that many natural linear dynamical systems ex- hibit chaos. In the past few years several open problems, some of which long-standing, have been solved, and a number of landmark results have been obtained.
From its very beginning, hypercyclicity has been at the crossroads of several areas of mathematics, by taking its examples and its techniques from various domains, and in turn its results have found applications and have motivated further research outside hypercyclicity.
To mention a few, hypercyclicity is connected with the following areas of mathematics: Operator Theory (e.g., through the invariant subspace prob- lem), Semigroups of operators and applications to PDEs, Dynamical systems (complex dynamics, non-linear dynamics, topological dynamics, ergodic the- ory).
Our purpose is to present a survey on recent advances and connections of hypercyclicity with other areas.
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Regular Holomorphic Functions on Complex Banach Lattices
Raymond Ryan.
National University of Ireland Galway, Ireland.
Abstract. Infinite dimensional holomorphy began on spaces like `2, where it was natural to imitate the approach taken in several complex vari- ables, so that a holomorphic function would be represented locally by a monomial expansionP
αcαzα. With the development of Functional Analyis on abstract spaces and the work of Gˆateaux and Fr´echet, this approach gave way to a representation by a power seriesP
nPn(z) of homogeneous polyno- mials. In the 1970’s, Boland and Dineen showed that monomial expansions could be used effectively in certain nuclear spaces with bases. However, in the case of a Banach space with unconditional basis, the monomial expansion might not even converge unconditionally at each point. In the 1980’s, Matos and Nachbin began to study the class of holomorphic functions for which the monomial expansion is unconditionally pointwise convergent. Grecu and the author showed that this approach could be interpreted in a coordinate-free setting by viewing the domain as a Banach lattice. We develop this idea and show how it is connected to the Fremlin tensor product for Banach lattices.
We use this approach to study some geometric properties of the domain of convergence of a power series and we explain the importance of the concepts of regularity and nuclearity.
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Geometry of Polynomials in Banach spaces and its applications to Bernstein and Markov inequalities
Juan B. Seoane Sep´ulveda
Universidad Complutense de Madrid. Spain.
Abstract. Let m, n ∈ N with m > n and consider the norm on R3 given by
kaxm+ bxn+ ckm,n = sup{|axm+ bxn+ c| : x ∈ [−1, 1]}
for every a, b, c ∈ R. In this talk we provide a full description of the extreme points of the corresponding unit balls and we also obtain, explicitly, the best possible constant Mm,n(x) in the inequality
(1) |p0(x)| ≤ Mm,n(x) · kpkm,n,
where x ∈ [−1, 1] is fixed and p is a trinomial as above. This answers a question to an old problem, first studied by Markov, for a large family of trinomials.
Analogously, if ∆ ⊂ R2 stands for the region enclosed by the triangle of vertices (0, 0), (0, 1) and (1, 0) (or simplex for short), one can also consider the space P(2∆) of 2-homogeneous polynomials on R2 endowed with the norm given by
kax2+ bxy + cy2k∆= sup{|ax2+ bxy + cy2| : (x, y) ∈ ∆}
for every a, b, c ∈ R. We provide a full description of the extreme points of the corresponding unit ball and, using this geometrical information, we find sharp Bernstein and Markov inequalities (as in equation (1)) for P(2∆) and also show that a classical inequality of Martin does not remain true for homogeneous polynomials on non symmetric convex bodies.
This talk is the result of several joint works with G. A. Mu˜noz-Fern´andez (Universidad Complutense de Madrid, Spain), Y. Sarantopoulos (National Technical University of Athens, Greece) and S. R´ev´esz (Hungarian Academy of Sciences. Budapest, Hungary).
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Linearization and Compactness
Ignacio Zalduendo.
Universidad Torcuato di Tella. Argentina.
Abstract. We review the idea of linearizations of function spaces, and then study two problems related to compactness properties: First, the prob- lem of characterizing when a Banach function space admits a Banach lin- earization in a natural way. Secondly, we study the relevance of compactness properties in linearizations, more precisely, the relation between compact- ness of a mapping, and compactness of its associated linear operator.
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SHORT TALKS
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Stability and instability of operators between spaces of continuous functions
Jes´us Araujo and Juan J. Font.
Universidad de Cantabria and Universitat Jaume I. Spain.
Abstract. Let > 0, and X, Y compact Hausdorff spaces. A continuous linear operator T : C(X) −→ C(Y ) is said to be -disjointness preserving if, given f, g ∈ C(X) with kf k∞= kgk∞= 1, f g ≡ 0 yields k(T f )(T g)k∞≤ .
Let C0 the set of all weighted composition maps from C(X) to C(Y ), and let C the set of all norm one -disjointness operators from C(X) to C(Y ).
For each , we consider
K() := sup{dist (T, C0) : T ∈ C}.
We study the following problems: How big can K() be? And how small?
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Multiplicative bijections of Lipschitz and smooth functions
Javier Cabello S´anchez.
Universidad de Extremadura. Spain.
Abstract. We prove the following results:
• If T : Lip (Y, I) → Lip (X, I) is a multiplicative bijection, where X and Y are complete metric spaces of finite diameter, then there is a Lipschitz homeomorfism τ : X → Y such that T f (x) = f (τ (x)) for any non-isolated x ∈ X.
• If T : Ck(Y, I) → Ck(X, I) is a multiplicative bijection, where X and Y are class-k smooth manifolds, then there is a class-k diffeomorfism τ : X → Y such that T f (x) = f (τ (x)) for any x ∈ X.
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Spectra of weighted algebras of holomorphic functions
Daniel Carando and Pablo Sevilla Peris.
Universidad de Buenos Aires (Argentina) and Universidad Polit´ecnica de Valencia (Spain)
Abstract. We consider weighted algebras of holomorphic functions on a Banach space. We determine conditions on a family of weights that as- sure that the corresponding weighted space is an algebra or has polynomial Schauder decompositions. We study the spectra of weighted algebras and endow them with an analytic structure. We also deal with composition op- erators and algebra homomorphisms, in particular to understand how their induced mappings act on the analytic structure of the spectrum. Moreover, a Banach-Stone type question is addressed.
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Hypercyclic and chaotic behaviour of C
0-semigroups generated by Ornstein-Uhlenbeck operators
Jos´e A. Conejero, Elisabetta Mangino and Alfredo Peris.
Universidad Polit´ecnica de Valencia, Universit`a del Salento and Universidad Polit´ecnica de Valencia.
Abstract. Let X be a separable infinite-dimensional Banach space. A strongly continuous semigroup of linear and continuous operators on L(X), i.e. a C0-semigroup, T = {Tt}t≥0 is said to be hypercyclic if {Ttx : t ≥ 0}
is dense in X for some x ∈ X. In addition, if it also has a dense set of periodic points, then it is said to be chaotic.
The investigation of hypercyclic and chaotic semigroups in this setting was initiated by Desch, Schappacher and Webb in [2]. They state some sufficient conditions on the infinitesimal generator of a semigroup to test whether it is hypercyclic and chaotic [2, Section 3]. These conditions have been generalized by Banasiak and Moszy´nski [1] and El Mourchid [3].
In this talk, we recall some facts concerning hypercyclic and chaotic semi- groups, and we use the foregoing conditions to estimate the hypercyclic and chaotic behavior of C0-semigroups generated by Ornstein-Uhlenbeck opera- tors of the form
n
X
i,j=1
qi,jDi,j+
n
X
i,j=1
bi,jxjDi+ αI, x ∈ Rn,
where Q = (qi,j) is a real, symmetric and positive definite matrix, B = (bi,j) is a non-zero real matrix, and α ∈ R.
References
[1] J. Banasiak and M. Moszy´nski. A generalization of Desch-Schappacher-Webb cri- teria for chaos. Discrete Contin. Dyn. Syst. 12(5), 959–972, 2005.
[2] W. Desch, W. Schappacher, and G.F. Webb. Hypercyclic and chaotic semigroups of linear operators. Ergodic Theory Dyn. Syst. 17(4), 793–819, 1997.
[3] S. El Mourchid. The imaginary point spectrum and hypercyclicity. Semigroup Forum 73(2), 313–316, 2005.
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A Banach-Stone theorem for D
∗spaces
Estibalitz Durand Cartagena and Jes´us A. Jaramillo Aguado.
Universidad Complutense de Madrid. Spain.
Abstract. For a metric space (X, d), we study the space D(X) of func- tions whose infinitesimal Lipschitz constant is uniformly bounded. More precisely, if for a function f : X → R we define the infinitesimal Lipschitz constant at a point x ∈ X as follows:
Lip f (x) = lim sup
y→x y6=x
|f (x) − f (y)|
d(x, y) , then
D(X) = {f : X −→ R, sup
x∈X
Lip f (x) < +∞}.
We compare D(X) with LIP(X) in terms of different properties regarding the geometry of X. We also obtain a Banach-Stone theorem in this context.
Namely, if D∗(X) denotes the space of bounded functions in D(X), then the natural Banach algebra structure on D∗(X) given by the norm
kf kD∗ = max{kf k∞, k Lip(f )k∞}.
determines the metric structure of X up to D−homeomorphisms for a com- plete metric space.
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Function of stabilization for dissipative dynamical systems
Arsen R. Dzhanoev, A.Loskutov and M.A.F.Sanju´an.
Moscow State University, Moscow (Russia), Moscow State University, Moscow (Russia) and Universidad Rey Juan Carlos (Spain).
Abstract. The standard Melnikov method for analyzing the onset of chaos in the vicinity of a separatrix is used to explore the possibility of suppression of chaos of a certain class of dynamical systems. For a given
dynamical system we apply an external perturbation, which we call the sta- bilizing perturbation, with the goal that after its action the chaos present in the system is suppressed. We apply this method to the nonlinear pendulum as a paradigm, and obtain some analytical expressions for the corresponding external perturbations that eliminate chaotic behavior.
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Extension of polynomials as a local property.
Maite Fern´andez-Unzueta and M. ´Angeles Prieto
CIMAT, Guanajuato (Mexico) and Universidad Complutense de Madrid (Spain)
Abstract. We will see that if a Banach space satisfies that every norm one polynomial defined on a subspace extends, then there are always exten- sions with uniformly bounded norm. We will discuss some consequences of this result.
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Tensor product representation of the (pre)dual of order continuous p-convex Banach lattices.
Irene Ferrando and Enrique A. S´anchez P´erez.
Instituto Universitario de Matem´atica Pura y Aplicada and Universidad Polit´ecnica de Valencia. Spain.
Abstract. Spaces of p-integrable functions with respect to a vector measure provide a general representation technique for p-convex Banach lattices. In the framework of the Lp spaces with respect to a vector m, our aim is to obtain a particular representations of the (pre)dual spaces of general Banach lattices. Our main tools are the duality properties of the integration map associated to the vector measure m; we provide suitable topologies for the tensor product of the space of q-integrable functions with respect to m (where p and q are conjugated real numbers) and the dual of the Banach space where m take its values. The main result asserts that under the assumption of compactness of the unit ball with respect to a par- ticular topology, the space Lp(m) can be written as the dual of a suitable normed space.
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On M -spaces and Banach spaces
Francisco Gallego Lupi´a˜nez.
Universidad Complutense de Madrid. Spain.
Abstract. We define in this paper the concept of C-space, related with M-spaces and Banach spaces. We obtain various properties on these spaces and propose some open problems.
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Flat sets, `
p-generating and fixing c
0in nonseparable setting
M. Fabian, A. Gonz´alez and V. Zizler.
Math. Institute, Czech Academy of Science, Universidad Polit´ecnica de Valencia and Math. Institute, Czech Academy of Science.
Abstract. In terms of the Kadec-Klee asymptotic smoothness we define p-flat (asymptotically p-flat) sets in Banach spaces and use these concepts in characterizing WCG (Asplund) spaces that are `p(ω1)-generated. As a byproduct we get an alternative proof of Rosenthal’s result on fixing c0(ω1) and on generation of subspaces of WCG spaces. In particular, we obtain that every subspace of `p(ω1), p ∈ (1, ∞] (resp. c0(ω1)) is `p(ω1)-generated (resp. c0(ω1) generated)
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The relationship between extreme homogeneous polynomials and multilinear forms on Hilbert spaces
Bogdan C. Grecu.
Queen’s University Belfast. UK.
Abstract. It is a well known fact that, in general, if Y is a subspace of the Banach space X then the extreme points of BY are not necessarily extreme in BX, but it is true that Ext(BX) ∩ Y ⊂ Ext(BY). Since the polarization constant for Hilbert spaces is 1, we can consider P(nH) as a closed subspace of L(nH). Using characterisation of extreme polynomials and multilinear forms, we show that for n = 2 (and any Hilbert space H) or 3 (and two-dimensional Hilbert spaces), an extreme point of BP(nH)remains extreme as considered as an element of BL(nH).
Joint work with G. A. Mu˜noz Fern´andez and J. B. Seoane Sep´ulveda, Universidad Complutense Madrid (Spain).
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On the moduli of Squareness
Antonio J. Guirao.
Universidad de Murcia. Spain.
Abstract. We introduce the notions of pointwise modulus of squareness and local modulus of squareness of a normed space X. This answers a question of C. Benitez, K. Przes lawski and D. Yost about the definition of a sensible localization of the modulus of squareness. Geometrical properties of the norm of X (Fr´echet smoothness, Gˆateaux smoothness, local uniform convexity or strict convexity) are characterized in terms of the behaviour of these moduli.
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Global Inversion on Length Spaces
Maribel Garrido, Olivia Gut´u and Jes´us A. Jaramillo.
Universidad Carlos III de Madrid, Universidad Aut´onoma del Estado de Hidalgo and Universidad Complutense de Madrid
Abstract. We give metric conditions for a local homeomorphism be- tween length spaces to be a covering map. An applications of this results to a Cartan Hadamard Theorem for infinite dimensional Finsler Manifols is also given.
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Zero sets of real polynomials
Petr H´ajek.
Czech Academy of Sciences. Czech Republic.
Abstract. On every real Banach space with a weak star separable dual there exists a homogeneous polynomial of any odd degree greater than one, having no infinite dimensional null space.
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Approximation numbers of nuclear and Hilbert-Schmidt multilinear forms defined on Hilbert spaces
Beatriz Hernando.
UNED. Spain.
Abstract. We deal with multilinear forms t defined on Hilbert spaces and we investigate the relationships between nuclearity of t (resp. to be of Hilbert-Schmidt type) and summability properties of certain approximation numbers of t.
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Functions locally dependent on finitely many coordinates
Petr H´ajek and Michal Johanis.
Mathematical Institute, Czech Academy of Science and Charles University Prague. Czech Republic.
Abstract. We give some results about functions locally dependent on finitely many coordinates, for example C∞-smooth UG approximations on c0(Γ).
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On bounded vector-valued holomorphic functions
L. Frerick, E. Jord´a and J. Wengenroth.
Universit¨at Trier, Universidad Polit´ecnica de Valencia and Universit´e de Li`ege.
Abstract. In this work we consider extensions of bounded vector-valued holomorphic functions defined on a unit ball U of a Banach space with values in another Banach space. These extensions are obtained from weak exten- sions. The results are based on the description of vector-valued functions as operators as a consequence of a Mujica’s theorem. As an application we prove a vector-valued version of Blaschke’s theorem.
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The geometry of the bidual of a separable Banach space
Sebasti´an Lajara.
Universidad de Castilla La Mancha, Spain.
Abstract.
A normed space (X, k · k) is said to be locally uniformly rotund (LUR) if, for every x ∈ X and every sequence (xn)n ⊂ X such that limnkxnk = kxk and limnkxn+ xk = 2kxk, we have limnkxn− xk = 0. A weakening of this property leads to the notion of midpoint locally uniform rotundity. The space X is said to be midpoint locally uniformly rotund (MLUR) if, for every x ∈ X and every sequence (xn)n ⊂ X such that limnkxn± xk = kxk, we have limnkxnk = 0.
It is well known that a Banach space is reflexive if the bidual space is locally uniformly rotund. The situation with MLUR norms is different.
Molt´o, Orihuela, Troyanski and Valdivia proved (Q. J. Math., 2001) that the James space J admits an equivalent norm k · k such that (J, k · k)∗∗
is MLUR. The aim of this talk is to show that there is a large class of separable Banach spaces with this property. Joint work with A. Pallares and S. Troyanski.
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Atomic decompositions for tensor products and polynomial spaces
S. Lassalle and D. Carando.
Universidad de Buenos Aires. Argentina.
Abstract. Atomic decompositions in Banach spaces are pairs of se- quences that allow reconstruction formulas that need not be unique, con- trary to what happens with basis. We study duality and reflexivity for atomic decompositions in Banach spaces extending to this setting the con- cepts of shrinking and boundedly complete bases.
We apply our results to study the existence of atomic decomposition for tensor products of Banach spaces and spaces of homogeneous polynomials.
If a Banach space X admits an atomic decomposition of a certain kind, we show that the symmetrized tensor product of the elements of the atomic decomposition provides an atomic decomposition for the symmetric tensor product Nn
s,µX, for any symmetric tensor norm µ. In addition, the recip- rocal statement is investigated.
Then we derive the existence of monomial atomic decompositions for cer- tain ideals of polynomials on X. Finally, the question of reflexivity of spaces of polynomials is also addressed.
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Orthogonally additive polynomials in `
pAlberto Ibort, Pablo Linares and Jos´e G. Llavona.
U. Carlos III (Spain), Universidad Complutense de Madrid (Spain) and Universidad Complutense de Madrid (Spain).
Abstract. We present a new proof of a Sundaresan’s result which shows that the space of orthogonally additive polynomials Po(k`p) is isometrically isomorphic to `p/p−k if k < p < ∞ and to `∞ if 1 ≤ p ≤ k.
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Essential norm of operators on weighted Bergman spaces of infinite order
Pablo Galindo and Mikael Lindstr¨om.
Universidad de Valencia (Spain) and Abo Akademi University (Finland).
Abstract. A formula for the essential norm of any operator between weighted Bergman spaces of infinite order will be discussed. We will also apply this formula to obtain or estimate essential norms of operators acting on Bloch type spaces and to differences of composition operators on some weighted Bergman spaces.
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Fr´ echet-Urysohn property for topological groups
Elena Mart´ın Peinador
Universidad Complutense de Madrid (Spain).
Abstract. We will present the following result: ”If the dual group of a metrizable abelian group has the Fr´echet-Urysohn (F-U) property, then it must be metrizable and locally compact”. As a consequence we will provide a big class of topological groups which are sequential but non F-U.
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Hankel Operators on Algebras of Analytic Functions
A. Miralles.
Universidad de Valencia (Spain).
Abstract. We prove that the Bourgain algebra of the polydisk algebra A(∆n) is A(∆n) itself and disprove the tightness of some algebras of analytic functions; in particular that of H∞(BE).
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Weak compactness and sigma-Asplund generated spaces
M. Fabian, V. Montesinos and V. Zizler.
Institut of Mathematics of the Czech Academy of Sciences (Czech Republic), Instituto Universitario de Matem´atica Pura y Aplicada,
Universidad Polit´ecnica de Valencia (Spain) and Institut of Mathematics of the Czech Academy of Sciences (Czech Republic).
Abstract. Sigma-Asplund generated Banach spaces are used to give new characterizations of subspaces of weakly compactly generated spaces and to prove some results around Radon-Nikod´ym compacta. We show, typically, that in the framework of weakly Lindel¨of determined Banach spaces, subspaces of weakly compactly generated spaces are the same as sigma-Asplund generated spaces. For this purpose, we study relationships between quantitative versions of Asplund property, dentability, differentia- bility, and of weak compactness in Banach spaces. As a consequence, we provide a functional-analytic proof of a result of Arvanitakis: A compact space is Eberlein if (and only if) it is simultaneously Corson and quasi- Radon-Nikod´ym.
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Unconditional constants in spaces of polynomials
Gustavo A. Mu˜noz Fern´andez.
Universidad Complutense de Madrid (Spain).
Abstract. If P is a homogeneous polynomial on Rn, given by P (x) = X
α∈Nn,|α|=n
aαxα, then we define the polynomial |P | by
|P |(x) = X
α∈Nn,|α|=n
|aα|xα.
Now if B is the unit ball of a Banach space in Rn and we define kP kB :=
sup{|P (x)| : x ∈ B}, then we investigate the inequality k|P |kB≤ CBkP kB,
providing sharp estimates on CB in some specific spaces of polynomials. We investigate the same problem for some spaces of non-homogeneous polyno- mials in one variable. It turns out that the constants CB correspond with the unconditional constants of the canonical bases of the above spaces. This research is part of a joint work with Bogdan C. Grecu and Juan B. Seoane Sep´ulveda.
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Hypercyclic convolution operators on Fr´ echet spaces of analytic functions
Daniel Carando, Ver´onica Dimant and Santiago Muro.
Universidad de Buenos Aires (Argentina), Universidad de San Andr´es (Argentina) and Universidad de Buenos Aires (Argentina).
Abstract. A result of Godefroy and Shapiro states that the convolution operators on the space of entire functions on Cn, which are not multiples of identity, are hypercyclic. Analogues of this result have appeared for some spaces of holomorphic functions on a Banach space. In this work, we define the space holomorphic functions associated to a sequence of spaces of poly- nomials and determine conditions on this sequence that assure hypercyclicity of convolution operators. Some known results come out as particular cases of this setting. We also consider holomorphic functions associated to mini- mal ideals of polynomials and to polynomials of the Schatten-von Neumann class.
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Bell inequalities: a long path from Einstein to Pisier
D. P´erez Garc´ıa, M.M. Wolf, C. Palazuelos, I. Villanueva and M.
Junge.
Universidad Complutense de Madrid (Spain), Max Planck Institut fuer Quantenoptik (Germany), Universidad Complutense de Madrid (Spain), Universidad Complutense de Madrid (Spain) and University
of Illinois at Urbana-Champaign (USA)
Abstract. Bell inequalities were originally designed to experimentally solve the metaphysical criticism of Quantum Mechanich made by Einstein, Podolski and Rosen in the 30’s. Nowadays they are a key tool in quantum information and quantum cryptography. We will show in this talk how tensor norms in Banach spaces, Banach algebras and operator spaces help to understand these inequalities. As a consequence we will show that there are tripartite quantum states which are arbitrarily robust against white noise, solving this way a long-standing open problem of B. Tsirelson.
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L
prepresentable functions on Banach spaces
Dami´an Pinasco.
Universidad de Buenos Aires (Argentina).
Abstract. D.Pinasco and I. Zalduendo [PZ] gave an integral represen- tation formula for holomorphic functions on separable Banach spaces and S.
Dineen and L. Nilsson [DN] worked on fully nuclear spaces.
We define the class of Lp - representable and ρ - representable functions on Banach spaces and study these function spaces. We also show that it is possible to give an alternative representation for integral polynomials over a Hilbert space H using a universal Wiener measure.
[DN] S. Dineen, L. Nilsson, Integral representation of holomorphic functions on fully nuclear spaces, Preprint.
[PZ] D. Pinasco and I: Zalduendo, Integral representation of holomorphic functions on Banach spaces. J. Math. Anal. Appl. 308(2005), 159-174.
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Winning tactics in a geometrical game
Anton´ın Proch´azka.
Charles University in Prague, Czech Republic and Universit´e Bordeaux 1, France.
Abstract. Let BX be a closed unit ball in a Banach space X. Let A be a class of subsets of BX such as open slices, closed slices, hyperplane sections, etc.
A point-A game in BX (first mentioned by Mal´y and Zelen´y for X = R2) is a two-player infinite game where Player I plays points xn in BX and Player II plays subsets An ∈ A according to the simple rules: xn+1 ∈ An and An 3 xn. Player II wins if (xn) is convergent, otherwise Player I wins.
Deville and Matheron have shown that X has the Radon-Nikod´ym property if and only if Player II has a winning strategy for the point-closed slice game in BX. We will show that Player II has a winning tactic for the point-closed slice game in BX provided X has the RNP. Also, we will see that Player II cannot have any winning tactic for the point-open slice game in BX.
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Biduality in weighted Banach spaces of holomorphic functions
Pilar Rueda.
Universidad de Valencia. Spain.
Abstract. This is a joint work with C. Boyd. Let U be an open sub- set of Rn. A weight v on U is any strictly positive real continuous func- tion on U which converges to 0 on the boundary of U . Let us consider the space Hv(U ) of all holomorphic functions f on U with the property that kf kv := supz∈Uv(z)|f (z)| is bounded, Hv(U ) endowed with the norm k kv is a Banach space. Let Hvo(U ) be the subspace of Hv(U ) of all holo- morphic functions f on U with the property that v|f | converges to 0 on the boundary of U . Consider the associated weights ˜v and ˜vo defined as
˜
v−1(z) = kδzkHv(U )∗ and ˜vo−1(z) = kδzkHvo(U )∗, where δz is the point eval- uation at z ∈ U . Our main result is that the bidual of Hvo(U ) is Hv(U ) if and only if Hvo(U ) is a M -ideal in Hv(U ) and the associated weights ˜v and
˜
vo coincide.
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Strictly singular inclusions into L
1+ L
∞V´ıctor Manuel S´anchez de los Reyes Universidad Complutense de Madrid. Spain.
Abstract. In this short talk it is given a complete characterization of the strict singularity and the disjoint strict singularity of the inclusions E ,→
L1+ L∞ for the class of rearrangement invariant function spaces E on the [0, ∞) interval. Their relationship is also analyzed. Suitable criteria are given involving the scale of order continuous weak Lp-spaces for 1 < p < ∞.
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On the real Plank problem and its applications
Y. Sarantopoulos, G. A. Mu˜noz-Fern´andez and J. B.
Seoane-Sep´ulveda.
National Technical University Athens (Greece), Universidad Complutense de Madrid (Spain) and Universidad Complutense de
Madrid (Spain).
Abstract. K. Ball [2] has proved the complex plank problem: if (xk)nk=1 is a sequence of norm 1 vectors in a complex Hilbert space (H, h·, ·i)), then there exists a unit vector x for which
|hx, xki| ≥ 1/√
n , k = 1, . . . , n .
In general, this result is not true on real Hilbert spaces. However, in special cases we prove that the same result holds true. As a consequence we give almost optimal lower bounds for the norms of homogeneous polynomials which are products of linear forms on real Hilbert spaces, see [1].
References
[1] J. Arias-de-Reyna, Gaussian variables, polynomials and permanents, Linear Algebra Appl. 285 (1998) 107–114.
[2] K.M. Ball, The complex plank problem, Bull. London Math. Soc. 33 (2001) 433–442.
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A functional Calculus for Quotient Bounded Operators
Sorin Mirel Stoian.
University of Petrosani (Romania).
Abstract. The classic functional calculus for the bounded operators on Banach spaces is naturally generalized for bounded operators on sequentially complete locally convex spaces, using the theory of holomorphic functions on locally convex spaces. This functional calculus is richer than Waelbroeck’s functional calculus.
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Twisting Schatten classes
Jes´us Su´arez.
Universidad de Extremadura. Spain.
Abstract. Kalton and Peck provided in [1] a method to construct twisted sums of `p with itself for 1 < p < ∞ , this is, a Banach space Zp contain- ing a non complemented copy of `p such that Zp/`p = `p. The technique employed in [1] is developed to “twist” spaces with unconditional bases so it cannot be extended to “twist” the Schatten classes Sp unless p = 2. We sketch a method to produce a twisted sum of the Schatten class Sp with itself for 1 < p < ∞. Additional properties of this new twisted sum like its B(H)-module structure will be considered.
References
[1] Kalton, N. J.; Peck, N. T., Twisted sums of sequence spaces and the three space problem, Trans. Amer. Math. Soc. 255 (1979), 1–30.
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Tensor Products of Besov Spaces and Applications
Tino Ullrich and Winfried Sickel.
University of Jena, Germany.
Abstract. We consider tensor products of Besov spaces, which yield Besov spaces with dominating mixed smoothness. Based on this we obtain spline-wavelet characterizations as well as new results for hyperbolic wavelet approximation.
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A class of hypercyclic Volterra composition operators
Andreas Weber and Gerd Herzog.
Universit¨at Karlsruhe (TH), Germany.
Abstract. It is known that the Volterra operator on Lp[0, 1], p ∈ [1, ∞) is cyclic but not supercyclic (cf. Gallardo-Guti´errez, Eva A., Montes-Rodr´ıguez, Alfonso, The Volterra operator is not supercyclic. Integral Equations Oper- ator Theory 50 (2004), no. 2, 211–216). In this talk, we show that certain Volterra composition operators are hypercyclic on the Fr´echet space of con- tinuous functions u : [0, 1) → R or C with u(0) = 0.
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