geometry, o s banach spacesgeometry operators series geometry, operators and series in banach spaces...

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GEOMETRY, OPERATORS AND SERIES INBANACH SPACES

Fernando Rambla Barreno

(as part of the Research Group FQM-257 (Univ. de Cadiz))

VIII Functional Analysis Network meeting (2012)

F. RAMBLA GEOMETRY, OPERATORS AND SERIES

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INTRODUCTION

The purpose of this talk is to review some of the resultsobtained by the group FQM-257, with an emphasis on themost recent ones.We come from a small university with a not-so-smalldegree in Mathematics! This implies a lot of young people.We will deal with several results in geometry, and a fewmore in operators and series.We are strongly open to collaborations!

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INTRODUCTION

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INTRODUCTION

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INTRODUCTION

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CONTENTS

1 GEOMETRYPathological renormings and noncomplete normedspacesIsometric reflection vectors, exposed faces and theseparable quotient problemTransitivity of the norm

2 OPERATORSPositive supercyclicityAnsari’s Theorem and a problem of S. Bourdon

3 SERIESSummability in the frame of Almost Convergence

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PATHOLOGICAL RENORMINGS AND NONCOMPLETE NORMED SPACESISOMETRIC REFLECTION VECTORS, EXPOSED FACES AND THE SQPTRANSITIVITY OF THE NORM

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PROBLEM

How small (or big) can be the set of functionals that do NOTattain their norm? (call it NNA)

THEOREM (JOINT WITH M. ACOSTA & R. ARON)

If X is an infinite-dimensional C(K ) or L1(µ) (with µ σ-finite)space, then NNA ∪ {0} contains an infinite-dimensional normedspace.

In the same paper, the following result was reproved:

THEOREM (M. ACOSTA & M. RUIZ GALAN)Every real Banach space can be renormed so that NNA isnondense.

We conjecture that every real Banach space can be renormedso that NNA is nowhere dense.THEOREM

Every real Banach space with separable dual can be renormedso that NNA is nowhere dense.

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We conjecture that every real Banach space can be renormedso that NNA is nowhere dense.

THEOREM

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THEOREM

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PROBLEM

What happens to rotundity and smoothness properties inBanach spaces if we drop the assumption of completeness?

THEOREM (JOINT WITH B. ZHENG)There exists a nonrotund Banach space with a rotund densemaximal subspace.

THEOREM (OP. CIT.)Every inf. dim. rotund Banach space can be renormed to benonrotund and contain a dense maximal rotund subspace.

THEOREM (OP. CIT.)There exists a nonsmooth Banach space with a smooth densemaximal subspace.

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PROBLEM

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Let X be a real normed space. A point x ∈ SX is an isometricreflection vector if there exists a closed maximal subspaceM ⊆ X such that X = L(e)⊕M and ‖λe + m‖ = ‖λe −m‖ forevery m ∈ M and every λ ∈ R. In this situation, there exists aunique e∗ ∈ SX∗ such that e∗(e) = 1 and e∗ is an isometricreflection vector in X ∗.The following is a fundamental result on these kind of vectors:

THEOREM (J. BECERRA & A. RODRIGUEZ-PALACIOS)If the set of isometric reflection vectors has nonempty interior inthe sphere, then X is a Hilbert space.

We gave an elementary proof of this fact in [5].

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More can be said, in this case concerning the “localization” ofthe concept.

THEOREM

In a Banach space, a point e is an isometric reflection vector ifand only if it is an isometric reflection vector in everytwo-dimensional subspace containing e.

Quite surprisingly, these concepts can be related ([9]) to the oldseparable quotient problem. Namely, given an isometricreflection vector e, consider the associated exposed maximalface Fe = (e∗)−1(1) ∩ BX . Then we have:

QUESTION (EQUIVALENT TO THE SEPARABLE QUOTIENT

PROBLEM)

Let X be a real Banach space. Can X be equivalentlyrenormed to have an isometric reflection vector e ∈ SX suchthat span(Fe − e) is a proper, dense subspace of ker(e∗)?

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THEOREM

PROBLEM)

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THEOREM

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Recall that a Banach space is almost transitive if for everyx , y ∈ SX and ε > 0 there exists a surjective linear isometryT : X → X satisfying ‖Tx − y‖ ≤ ε. It is even transitive if wecan take ε = 0.

CONJECTURE (BANACH-MAZUR, 1932)Every transitive separable Banach space is a Hilbert space.

THEOREM (MAZUR)Every transitive separable Banach space is smooth.

THEOREM

If a Banach space X is transitive, has normal structure and theexposed faces of the unit ball are pairwise disjoint then X isrotund.

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THEOREM

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THEOREM (GREIM & RAJAGOPALAN / FQM-257, K.KAWAMURA)

No infinite-dimensional real C0(L) space is almost transitive.There exists an infinite-dimensional complex C0(L) space whichis almost transitive.

PROBLEM

If a Banach space is isomorphic to a Hilbert space and almosttransitive, is it a Hilbert space?

PROBLEM

Does every Banach space has a renorming whose group ofisometries is maximal?

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POSITIVE SUPERCYCLICITYANSARI’S THEOREM AND A PROBLEM OF S. BOURDON

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Let X be a topological space and f : X → X a continuousmapping. Then f is said to be hypercyclic if there exists x ∈ X ,called a hypercyclic point for f , such that the orbit{f n(x) : n ∈ N} is dense in X . If X has a richer structure, let ussay it is a topological vector space, then f is said to be

supercyclic if there exists x ∈ X , called a supercyclicpoint for f , such that {λf n(x) : n ∈ N, λ ∈ K} is dense in X .cyclic if there exists x ∈ X , called a cyclic point for f , suchthat span {f n(x) : n ∈ N} is dense in X .

There are weak versions of these notions, defined using weakdensity. The classical invariant subspace problem can berephrased as: In a given separable Banach space, does everyoperator have a noncyclic, nonnull vector?

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THEOREM (JOINT WITH V. MULLER)Let X be a Banach space and T : X → X an operator. If x is ahypercyclic vector for T then for every λ ∈ SC, x is a hypercyclicvector for λT .

THEOREM

Let X be a Banach space and T : X → X an operator withweakly supercyclic vector x ∈ X. If T ∗ has no eigenvalues, thenx is also weakly positive supercyclic for T , i.e.{λT nx : λ ≥ 0,n ∈ N} is weakly dense in X.

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THEOREM (JOINT WITH V. MULLER)Let X be a Banach space and T : X → X an operator. If x is ahypercyclic vector for T then for every λ ∈ SC, x is a hypercyclicvector for λT .

THEOREM

Let X be a Banach space and T : X → X an operator withweakly supercyclic vector x ∈ X. If T ∗ has no eigenvalues, thenx is also weakly positive supercyclic for T , i.e.{λT nx : λ ≥ 0,n ∈ N} is weakly dense in X.

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Whenever X is a perfect Hausdorff topological space,f : X → X is continuous and x ∈ Hyp(f ) \ Hyp(f 2), there existsa decomposition of X into three sets with certain properties.This is a result of Bourdon which, in the case n = 2, implies thefollowing famous result of S. Ansari:

THEOREM (S. ANSARI)

If T is a hypercyclic operator in a Banach space, then T n is alsohypercyclic for every n ∈ N. Moreover, all powers share thesame hypercyclic vectors.

Bourdon suggested finding a similar decomposition in the caseof arbitrary n. This was accomplished in [12]:

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THEOREM (S. ANSARI)

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THEOREM (S. ANSARI)

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THEOREM (JOINT WITH K.-G. GROSSE-ERDMANN)Let X be a perfect, Hausdorff topological space and f : X → Xcontinuous. If n ∈ N and x ∈ Hyp(f ), the following areequivalent:

{f n(x) : x ∈ X} is not dense in X.There exist k ∈ N with k > 1 and k |n, and a partition ofHyp(f ) into sets D1, . . . ,Dk that are invariant under f k .

From this result it is not difficult to obtain Bourdon’s result (inthe case n = 2) and also to give a simple proof of the followinggeneralization of Ansari’s Theorem:

THEOREM (J. WENGENROTH)If T is a hypercyclic operator in a topological vector space, thenT n is also hypercyclic for every n ∈ N. Moreover, all powersshare the same hypercyclic vectors.

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SERIESSUMMABILITY IN THE FRAME OF ALMOST CONVERGENCE

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DEFINITION (BANACH, 1932)A Banach limit is a linear function ϕ : `∞ → R such that:

ϕ(x) ≥ 0 if x ≥ 0.ϕ((xn)n) = ϕ((xn)n+1) for every x ∈ `∞.ϕ(e) = 1.

A bounded sequence (xn)n ⊆ `∞ is almost convergent(Lorentz, 1948) if there exists y ∈ `∞ such that ϕ(xn)→ ϕ(y)for every Banach limit ϕ. We denote this by AC − lım xn = x . Itturns out that these sequences can be characterized verynaturally:

THEOREM (LORENTZ)

A sequence (xn)n in a Banach space X is almost convergent to

x ∈ X if and only if xi+xi+1+···+xi+j−1j

j→ x uniformly on i ∈ N.

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THEOREM (LORENTZ)

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Using this characterization we proved a new, slightly strongerversion of the classical Orlicz-Pettis theorem:

THEOREM

Let X be a Banach space and ξ =∑∞

n=1 xn a series in X suchthat w − AC

∑n∈M xn exists for every M ⊆ N. Then ξ is

absolutely convergent.

It seems logical to extend the notion of Banach limit to generalnormed spaces by using Lorentz’s characterization. This is aquestion we are currently studying.

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Using this characterization we proved a new, slightly strongerversion of the classical Orlicz-Pettis theorem:

THEOREM

Let X be a Banach space and ξ =∑∞

n=1 xn a series in X suchthat w − AC

∑n∈M xn exists for every M ⊆ N. Then ξ is

absolutely convergent.

It seems logical to extend the notion of Banach limit to generalnormed spaces by using Lorentz’s characterization. This is aquestion we are currently studying.

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Some other topics:

Statistical convergence. Asymptotic Chebyshev centers.A Bishop-Phelps-Bollobas modulus. Effect Algebras.Diameter of functions and Banach-Stone theorems.Lineability.

Current members of the group: Leon Saavedra, F.; ArmarioMigueles, R.; Garcıa Pacheco, F. J.; Listan Garcıa, M. C.;Moreno Pulido, S.; Nicasio Llach, M.; Perez Fernandez, F. J.;Piqueras Lerena, A.; Rambla Barreno, F.; Romero de la Rosa,M. P.; Sala Perez, A.; Suarez Aleman, C. O.; Tamayo Rivera, M.y Villegas Vallecillos, M.

We are strongly open to collaborations!Thank you for your attention!

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Some other topics:

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ACOSTA, M. D.; AIZPURU, A. ; ARON, R. M.;GARCIA-PACHECO, F. J., Functionals that do not attain theirnorm, Bull. Belg. Math. Soc. Simon Stevin 14 (2007), no. 3,407–418.

ACOSTA, M. D.; RUIZ GALAN, M., New characterizations ofthe reflexivity in terms of the set of norm attainingfunctionals. (English summary), Canad. Math. Bull. 41(1998), no. 3, 279–289.

AIZPURU, A.; ARMARIO, R.; GARCIA-PACHECO, F. J;PEREZ-FERNANDEZ, F. J., Banach limits and uniformalmost summability, J. Math. Anal. Appl. 379 (2011), no. 1,82–90.

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AIZPURU, A.; ARMARIO, R.; PEREZ-FERNANDEZ, F. J.,Almost summability and unconditionally Cauchy series,Bull. Belg. Math. Soc. Simon Stevin 15 (2008), no. 4,635–644.

AIZPURU, A.; GARCIA-PACHECO, Rotundity in transitiveand separable Banach spaces, Quaest. Math. 30 (2007),85–96.

AIZPURU, A.; GARCIA-PACHECO, F. J; RAMBLA, F.,Isometric reflection vectors in Banach spaces, J. Math.Anal. Appl. 299 (2004), no. 1, 40–48.

AIZPURU, A.; RAMBLA, F., Diameter preserving linearbijections and C0(L) spaces, Bull. Belg. Math. Soc. SimonStevin 17 (2010), no. 2, 377–383.

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GARCIA-PACHECO, F. J.; RAMBLA-BARRENO, F.;SEOANE-SEPULVEDA, J. B., Q-linear functions, functionswith dense graph, and everywhere surjectivity, Math.Scand. 102 (2008), no. 1, 156–160.

GARCIA-PACHECO, F. J., Geometry of isometric reflectionvectors, Math. Slovaca 61 (2011), no. 5, 807–816.

GARCIA-PACHECO, F. J., Nowhere density of the set ofnon-norm-attaining functionals, Operator algebras, operatortheory and applications, 167–172, Oper. Theory Adv. Appl.195, Birkhuser Verlag, Basel, 2010.

GARCIA-PACHECO, F. J., Isometric reflection in twodimensions and dual L1-structures, to appear in Bull.Korean Math. Soc.

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GROSSE-ERDMANN, K.-G.; LEON-SAAVEDRA, F.;PIQUERAS-LERENA, A., The iterates of a map with denseorbit, Acta Sci. Math. (Szeged) 74 (2008), no. 1-2, 245–257.

GREIM, P.; RAJAGOPALAN, M., Almost transitivity in C0(L),Math. Proc. Cambridge Philos. Soc. 121 (1997), no. 1,75–80.

GARCIA-PACHECO, F. J.; ZHENG, B., Geometric Propertieson non-complete spaces, Quaest. Math. 34 (2011),489–511.

KAWAMURA, KAZUHIRO, On a conjecture of Wood, Glasg.Math. J. 47 (2005), no. 1, 1–5.

LEON-SAAVEDRA, F.; MULLER, V., Rotations of hypercyclicand supercyclic operators, Integral Equations OperatorTheory 50 (2004), no. 3, 385–391.

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LEON-SAAVEDRA, F.; PIQUERAS-LERENA, A., On weakpositive supercyclicity, Israel J. Math. 167 (2008), 303–313.

LEON-SAAVEDRA, F.; PIQUERAS-LERENA, A., Positivity inthe theory of supercyclic operators, Perspectives inoperator theory, 221–232, Banach Center Publ. 75, PolishAcad. Sci., Warsaw (2007).

RAMBLA, F., A counterexample to Wood’s conjecture., J.Math. Anal. Appl. 317 (2006), no. 2, 659667.

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