Special Session 13 Non-linear Analysis in Banach Spaces Convention Center August 8 & 9, 2013

Transcription

1 Special Session 13 Non-linear Analysis in Banach Spaces Convention Center August 8 & 9, 2013

2 Geraldo Botelho* Faculdade de Matemática, Universidade Federal de Uberlândia, Uberlândia, , Brazil. On compact extensions of multilinear operators on Banach spaces. Using the generalization of Kakutani s precompactness lemma due to Mujica, we give a sufficient condition for Nicodemi extensions of compact multilinear operators between Banach spaces to be compact as well. An application of this result to the isometric/isomorphic theory of spaces of compact multilinear operators is provided. This is a joint work with Kuo Po Ling. (Received April 26, 2013) 1

3 Daniel Carando, Pab. I, Ciudad Universitaria (1428), Buenos Aires, Argentina, Daniel Galicer, Pab. I, Ciudad Universitaria (1428), Buenos Aires, Argentina, and Damián Pinasco* Saenz Valiente 1010 (1428), Buenos Aires, Argentina. Distance geometry of convex bodies. Let K R n be a compact set endowed with the metric d α (x, y) = x y α 2, where 0 < α < 1. A classical result of Schoenberg and Von Neumann asserts that there exist a minimum r for which the metric space (K, d α ) may be isometrically embedded on a sphere of radius r in a Hilbert space. In this talk we will discuss estimates of these radii for some centrally symmetric convex bodies. For this purpose, we study the energy integral sup x y 2α dµ(x)dµ(y), K K where the supremum runs over all finite signed Borel measures on K of unit mass. (Received May 14, 2013) 2

4 Gilles Pisier* Quantum Expanders and Geometry of Operator Spaces. We show that there are well separated families of quantum expanders with asymptotically the maximal cardinality allowed by a known upper bound. This has geometric applications to the local theory of operator spaces. This allows us to provide sharp estimates for the growth of the multiplicity of M N -spaces needed to represent (up to a constant C > 1) the M N -version of the n-dimensional operator Hilbert space OH n as a direct sum of copies of M N. We show that, when C is close to 1, this multiplicity grows as exp βnn 2 for some constant β > 0. The main idea is to identify quantum expanders with smooth points on the matricial analogue of the unit sphere. This generalizes to operator spaces a classical geometric result on n-dimensional Hilbert space (corresponding to N = 1). Our work strongly suggests to further study a certain class of operator spaces that we call matricially subgaussian. (Received March 29, 2013) 3

5 William B. Johnson* Department of Mathematics, Texas A&M University, College Station, TX Subspaces of L p that embed into L p (µ) with µ finite. 40 years ago, Enflo and Rosenthal proved that l p (ℵ 1 ), 1 < p < 2, does not (isomorphically) embed into L p (µ) with µ a finite measure. Schechtman and I complement that theorem by proving that if X is a subspace of an L p space, 1 < p < 2, and l p (ℵ 1 ) does not embed into X, then X embeds into L p (µ) for some finite measure µ. This work is an outgrowth of an unsuccessful attempt to solve a problem left open by Enflo and Rosenthal; namely, whether L p (µ) can have an unconditional basis when 1 < p 2 <, the measure µ is finite, and the density character of L p (µ) is ℵ 1. If the answer is yes, then the results of Enflo and Rosenthal would show that L p ({ 1, 1}) 2ℵ 0 ) has an unconditional basis is undecidable. (This work is joint with Gideon Schechtman.) (Received May 14, 2013) 4

6 Rubén A. Martínez-Avendaño* Some remarks on subspace-hypercyclicity. Let T be a bounded linear operator on a Banach space X and let M be a nonzero subspace of X. If x X, we define the orbit of T under x as the set Orb(T, x) := {x, T x, T 2 x, T 3 x,... }. The operator T is called subspace hypercyclic for M if there exists a nonzero vector x X such that the intersection of M with Orb(T, x) is dense in M. If M = X, this is just the usual concept of hypercyclicity. In this talk we will introduce some basic facts about subspace hypercyclic operators, we will show some examples, and we will mention some open questions about subspace hypercyclic operators. (Received May 14, 2013) 5

7 Mary L Lourenco* (lourencomarylilian@gmail.com), University of São Paulo, São Paulo, Sao Paulo. Boundaries for Algebras of Analytic Functions on some Banach Spaces. For a Banach space X, let A be the Banach algebra of all complex valued functions defined on the closed unity ball B X that are uniformly continuous on B X and holomorphic on the interior of B X.. In this talk, we will focus in norming subset of B X, i.e., boundaries for the algebra A. In particular, we will show that the torus is a boundary for A where X is an l -sum of infinitely many Banch spaces. (Received April 11, 2013) 6

8 Richard Aron, Daniel Carando, Ted Gamelin, Silvia Lassalle* and Manuel Maestre. Cluster sets of analytic functions on Banach spaces. We present joint work with Richard Aron, Daniel Carando, Ted Gamelin and Manuel Maestre. Let B be the open unit ball of a complex Banach space X and B the closed unit ball of X, the bidual of X. Fixed f a holomorphic function bounded on B and z B, the cluster set of f at z, Cl r (f, z), is defined to be the set of all limits of values of f along nets in B converging weak-star to z. We study uniform algebras of bounded analytic functions on B in relation to cluster sets. We prove several cluster value theorems, relating cluster sets of a function to its range on the fibers of the spectrum of the algebra. These lead to weak versions of the corona theorem for l 2 and for c 0. We also address the analogous situation for H b (X), the Fréchet algebra of all entire functions which are bounded on bounded sets of X. (Received May 14, 2013) 7

9 Lawrence A Harris* (larry@ms.uky.edu), Mathematics Department, 715 Patterson Office Tower, University of Kentucky, Lexington, KY Markov theorems for polynomials on real normed spaces. In this talk we consider sharp upper bounds on the norm of the n th order Gâteaux and Fréchet derivatives for polynomials with a given bound on the closed unit ball of real normed linear spaces. We show that V. A. Markov s classical bounds continue to hold for the Gâteaux derivatives and that better estimates hold that depend on the value of polynomial at the point where the derivative is taken and at the origin. These results are established by a new technique that uses an explicit formula for bivariate Lagrange polynomials for certain nodes in the plane and a corresponding Christoffel-Darboux formula. Bounds on the derivatives are obtained directly from bounds on the derivatives of the Lagrange polynomials. Finding sharp bounds on the Fréchet derivatives seems more difficult. We discuss some results for certain cases. (Received February 19, 2013) 8

10 Santiago Muro* Damián Pinasco and Martín Savransky. Mixing operators on spaces of entire functions on a Banach space. Preliminary report. In this talk we analyze the dynamics of two classes of operators defined on spaces of entire functions of bounded type on a Banach space. First we deal with convolution operators. A theorem of Godefroy and Shapiro [3] states that non-trivial convolution operators on H(C n ) are hypercyclic. This theorem was improved in [2], where it is shown that non-trivial convolution operators are frequently hypercyclic and that there are frequently hypercyclic entire functions of exponential growth. We show that the result from [2] may be extended to some spaces of holomorphic functions on Banach spaces. Then we study the behavior of some non-convolution operators. In [1] it is proved that the operator T f(z) = f (az +b) with a, b C is hypercyclic on H(C) for a 1. We investigate the dynamics of analogues of this operator, defined on some spaces of holomorphic functions on a Banach space. [1] R. Aron and D. Markose. On universal functions.j. Korean Math. Soc., [2] A. Bonilla and K. Grosse-Erdmann. On a theorem of Godefroy and Shapiro.Integral Equations Operator Theory, [3] G. Godefroy and J. Shapiro. Operators with dense, invariant, cyclic vector manifolds.j. Funct. Anal., (Received May 15, 2013) 9

11 Thomas B Schlumprecht* (schlump@math.tamu.edu), Department of Mathematics, Texas A&M University, College Station, TX On Zippin s Embedding Theorem. We present a new proof of Zippin s Embedding Theorem, which states that every Banach space X which has a separable dual embeds into a space Z with shrinking basis and every separable reflexive Banach space embeds into a space Z with shrinking and boundedly complete bases. We also obtain a Theorem due to Johnson and Zheng which characterizes the spaces X for which the basis of Z can be chosen unconditional. (Received April 02, 2013) 10

12 Daniel Eric Galicer* Verónica Dimant and Ricardo García. Geometry of integral polynomials, M-ideals and unique norm preserving extensions. We use the Aron-Berner extension to prove that the set of extreme points of the unit ball of the space of integral k- homogeneous polynomials over a real Banach space X is {±φ k : φ X, φ = 1}. With this description we show that, for real Banach spaces X and Y, if X is a non trivial M-ideal in Y, then k,s ε k,s X (the k-th symmetric tensor product of X endowed with the injective symmetric tensor norm) is never an M-ideal in k,s ε k,s Y. 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 ε k X (the k-th tensor product of X endowed with the injective tensor norm) is an M-ideal in k ε k Y. Nevertheless, if X is also Asplund, we prove that every integral k-homogeneous polynomial in X has a unique extension to Y that preserves the integral norm. Other applications to the metric and isomorphic theory of symmetric tensor products and polynomial ideals are also given. (Received May 14, 2013) 11

13 Daniel Carando* Andreas Defant and Pablo Sevilla-Peris. Bohr s absolute convergence problem for H p -Dirichlet series in Banach spaces. The Bohr-Bohnenblust-Hille Theorem states that the width of the strip in the complex plane on which an ordinary Dirichlet series n a nn s converges uniformly but not absolutely is less than or equal to 1/2, and this estimate is optimal. Equivalently, the supremum of the absolute convergence abscissas of all Dirichlet series in the Hardy space H equals 1/2. By a surprising fact of Bayart the same result holds true if H is replaced by any Hardy space H p, 1 p <, of Dirichlet series. For Dirichlet series with coefficients in a Banach space X the maximal width of Bohr s strips depend on the geometry of X; Defant, García, Maestre and Pérez-García proved that such maximal width equal 1 1/Cot(X), where Cot(X) denotes the optimal cotype of X. Equivalently, the supremum over the absolute convergence abscissas of all Dirichlet series in the vector-valued Hardy space H (X) equals 1 1/Cot(X). In this talk we will revisit these facts and present our contribution: the last result remains true if H (X) is replaced by the larger class H p (X), 1 p <. This is a joint work with Andreas Defant and Pablo Sevilla-Peris (Received May 15, 2013) 12

14 Maria D. Acosta, Pablo Galindo and Luiza A. Moraes* Tauberian Polynomials on Banach Spaces. We say that a continuous n-homogeneous polynomial P : E F is Tauberian if P (E \E) F \F, where P denotes the Aron-Berner extension of P ( which is a polynomial from E into F ). In case of a continuous linear operator T : E F, this condition means that T 1 (F ) E (where T denotes a bi-transpose of T ). Continuous linear operators satisfying this condition have been studied by N. Kalton and A. Wilansky, who refered to them as Tauberian operators. In this talk we will see how some of the ideas behind the notion of Tauberian operator can be seen from a multilinear or polynomial point of view and to which extent results valid for Tauberian operators still hold in such non linear setting. When loosing linearity new difficulties arise and different tools are required. References [1] M. D. Acosta, P. Galindo e L. A. Moraes, Tauberian polynomials, pre-print. [2] N. J. Kalton e A. Wilansky, Tauberian operators on Banach spaces, Proc. Amer. Math. Soc. 57 (1976) (Received May 04, 2013) 13

15 Daniel Marinho Pellegrino* Departamento de Matemática, UFPB, Joao Pessoa, Paraiba , Brazil. The Bohnenblust Hille inequalities: past and present. We present various recent results, of several authors, on the estimates of the constants of the polynomial and multilinear (complex and real) Bohnenblust Hille inequalities. We also intend to give an overview of the subject since the publication of the original paper of H.F. Bohnenblust and E. Hille, in (Received April 14, 2013) 14