We report on the meeting Operators in Banach spaces recently held in Castro Urdiales as a homage to Pietro Aiena, and we collect the questions proposed by the participants during the Open Problems Session.Ministerio de Ciencia e Innovación (MICINN) de España. Grant MTM2011-2653 y MTM2010-20190.peerReviewe

Bermúdez, T.

González, M.

Pello, J.

Extracta Mathematicae

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Institutional Repository University of Extremadura

Several open problems in operator theory

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E extracta mathematicae Vol. 28, Núm. 2, 149 – 156 (2013)Several Open Problems in Operator TheoryT. Bermúdez, M. González, J. Pello ∗Departamento de Análisis Matemático, Universidad de La Laguna38271 La Laguna (Tenerife), Spain, tbermude@ull.esDepartamento de Matemáticas, Universidad de Cantabria39071 Santander, Spain, manuel.gonzalez@unican.esDepartamento de Matemática Aplicada, Universidad Rey Juan Carlos28933 Móstoles (Madrid), Spain, javier.pello@urjc.esPresented by Manuel González Received November 28, 2013Abstract : We report on the meeting Operators in Banach spaces recently held in CastroUrdiales as a homage to Pietro Aiena, and we collect the questions proposed by the partic-ipants during the Open Problems Session.Key words: operator theory, open problems.AMS Subject Class. (2010): 47A11, 47A53, 47A55.The conference “Operators on Banach spaces” was held in Castro Urdiales,a town located in the west of Cantabria (a region in the north of Spain), duringthe second week of June 2013. It was the tenth edition of a series of meetingsabout Banach spaces and operator theory organised by the Universities ofExtremadura and Cantabria.Operator theory is the research field of Pietro Aiena, to whom this meetingpaid a well-deserved homage. He began his research career with several paperson Fredholm theory around 1980, and soon he concentrated his efforts in thestudy of the spectral theory of multipliers. Around 1990 he discovered theusefulness of local spectral theory, and he contributed to its developmentand successfully applied it to the study of the spectrum of special classes ofoperators on Banach spaces. A good part of this research until 2002 can befound in his monograph [5], and he has continued doing research on this topicsince then. He was the supervisor of 7 Ph. D. Theses, and MathSciNet showsmore than 90 items authored by him.The invited speakers of the conference were José Bonet (Valencia), CristinaCâmara (Lisboa), Robin Harte (Dublin), Francisco L. Hernández (Madrid),∗The first author is supported by MICINN (Spain), Grant MTM2011-26538. The othertwo authors are supported by MICINN (Spain), MTM2010-20190.149150 t. bermúdez, m. gonzález, j. pelloTeresa Malheiro (Braga), Martin Mathieu (Belfast), Alfonso Montes Rodŕı-guez (Sevilla), Vladimı́r Müller (Praha) and Florian Vasilescu (Lille). Therewere also several sessions of short communications and plenty of discussiontime. Among the attendants we could meet many friends of Pietro Aiena,including mathematicians from Canada, México, Japan, Poland, Germany,Belgium, France, Italy and Spain. Moreover Jesús Ramón Guillén Ruiz andPedro Leonardo Peña Duarte, from Universidad de los Andes (Mérida), rep-resented the mathematicians from Venezuela whose Ph. D. Theses have beensupervised by Pietro Aiena.The venue of the conference was the Centro Cultural y de Congresos “LaResidencia”, close to the fishing port of Castro Urdiales, which is the cus-tomary venue of the activities of the Centro Internacional de EncuentrosMatemáticos (CIEM), a joint initiative of the University of Cantabria andthe City Council of Castro Urdiales.The organisers of the conference were Jesús M. F. Castillo (Badajoz),Manuel González (Santander) and Javier Pello (Madrid). They acknowledgefinancial support from the Universities of Cantabria and Palermo, and fromthe town council of Castro Urdiales.For additional information we refer to the official web page of the confer-ence:http://www.ciem.unican.es/encuentros/banach/2013/Open problems sessionHere we collect the questions that were posed and discussed during theOpen Problems Session of the conference. We thank the participants for theirinterest and enthusiasm, which provided a stimulating working environment.1. Hypercyclic and chaotic operators on Fréchet spacesJosé Bonet (Universidad Politécnica de Valencia)We begin by introducing the concepts involved in this part. We refer to[13] for additional information.Definition 1. Let T be an operator acting on a Fréchet space X.The operator T is called hypercyclic if there exists a point x ∈ X suchthat the set {Tnx : n ∈ N} is dense in X.A point x ∈ X is called a periodic point of T if there is some n ≥ 1 suchthat Tnx = x.open problems in operator theory 151The operator T is called chaotic if it is hypercyclic and has a dense set ofperiodic points.Hypercyclic operators behave in a different way when they are defined ona Banach space or on a non-normable Fréchet space in the following sense:1. Let X be a Banach space. Given an operator T ∈ L(X) and a complexnumber λ such that ∥λT∥ ≤ 1, obviously λT is not hypercyclic.2. Let D be the differentiation operator, which is defined on the space ofentire functions H(C) by Df = f ′. Then λD is hypercyclic for anyλ ∈ C \ {0}.Question 1. Suppose that E is a non-normable, separable Fréchet space.Is there T ∈ L(E) such that λT is hypercyclic for any λ ∈ C \ {0}?Some partial results were given by Frerick and Peris.Proposition 1. [10] Every separable Fréchet space E without a contin-uous norm admits a continuous linear operator T ∈ L(E) such that λT ishypercyclic for each λ ̸= 0.Question 2. Let E be a norm-normable and separable Fréchet space. Isthere an infinite-dimensional subspace A ⊂ L(E) such that T is hypercyclicfor any non-zero operator T ∈ A?Aron observed that the answer of Question 2 is positive for the Fréchetspaces H(C) and C∞(R), as follows from results by Godefroy and Shapiro [11].It also follows from results of Bès and Conejero [6] that this is the case for theFréchet space ω of all complex sequences.Bonet, Mart́ınez and Peris [8] proved that there exists a Banach space Xthat admits no chaotic operator.Question 3. Which Fréchet spaces admit a chaotic operator?Recently, de la Rosa, Frerick, Grivaux and Peris [9] proved that Banachspaces with an unconditional basis and Fréchet spaces with an unconditionalbasis and a continuous norm admit a chaotic operator; thus giving a partialanswer to Question 3.Some of these questions appear in [7].152 t. bermúdez, m. gonzález, j. pello2. A question on the rotundity of ballsKazimierz Goebel (Maria Curie-Sklodowska University)The following questions have over 20 years of history. They were firstasked in 1989, at the Fixed Point and Applications Conference in Marseille,Luminy.Let (X, ∥·∥) be a Banach space.The standard method of measuring “the rotundity ” of the unit ball in Xis through the modulus of convexity δX : [0, 2] → [0, 1] of XδX(ε) := inf{1−∥∥∥∥x+ y2∥∥∥∥ : ∥x∥ ≤ 1, ∥y∥ ≤ 1, ∥x− y∥ ≥ ε}and the characteristic of convexityε0(X) := sup{ε : δX(ε) = 0} .The modulus of convexity has “two-dimensional character”, meaning thatδX(ε) = inf{δE(ε) : E ⊂ X , dimE = 2} .It is known that the Hilbert space H is the most rotund space among allBanach spaces X, in the sense thatδX(ε) ≤ δH(ε) = δE2(ε) = 1−√1− ε24,where E2 is the two-dimensional Euclidean space.Now, fix a ∈ [0, 2) and consider the class Ea of all two-dimensional spaces(E, ∥·∥) having ε0(E) = a. Which of these spaces is the most rotund?It can be formulated in the following questions:Question 4. Given ε ∈ [a, 2), what is sup{δE(ε) : E ∈ Ea}?Question 5. Does there exist a space Ea ∈ Ea such that δE(ε) ≤ δEa(ε)for all E ∈ Ea?Question 6. If the answer to the above is positive, is such a space Ea insome sense unique?For additional information we refer to [12, page 35].open problems in operator theory 1533. Spectra of the restriction and quotient maps associatedto a particular subspaceRobin Harte (Trinity College Dublin)Let X be a complex Banach space and let T ∈ L(X) be a bounded opera-tor. We consider the point spectrum and the surjective spectrum of T whichare defined as follows:π(T ) = {λ ∈ C : λ− T is not injective} ,π′(T ) = {λ ∈ C : λ− T is not surjective} ,and the subspaceY := EX(T ) :=∑λ∈C(λ− T )−∞(0) =∪0̸=p∈Pp(T )−1(0) ,where P is the set of all polynomials.The set Y is a linear subspace not necessarily closed but could be dense.Let us denote by TY : Y → Y and T ′Y : X/Y → X/Y the restriction and thequotient operator associated to T and Y .Question 7. Are the sets π(T ), π(TY ) and π′(TY ) equal?Question 8. Is π(T ′Y ) = ∅?Question 9. What can we say about π(T ) and π′(T ) when Y is dense?4. Spectral characterisations of Jordan homomorphismsMartin Mathieu (Queen’s University Belfast)Let A and B be semisimple Banach algebras. A linear operator T : A → Bis called1. spectrally bounded if there exists M ≥ 0 such that for all a ∈ Ar(Ta) ≤ Mr(a) ,where r(·) denotes the spectral radius;2. a spectral isometry if for all a ∈ A it holds that r(Ta) = r(a);3. unital if T1 = 1.154 t. bermúdez, m. gonzález, j. pelloKaplansky’s Question (1970) : If T is unital and surjective such thatσ(Ta) = σ(a) for all a ∈ A, where σ(·) denotes the spectrum, does it followthat T is a “Jordan homomorphism”, that is, T (a2) = T (a)2 for all a ∈ A?Mathieu’s Conjecture (2001) : Let A and B be unital C∗-algebras. IfT is a unital surjective spectral isometry, then T is a Jordan homomorphism.Proposition 2. ([15]) Let T : Mn(C) → Mn(C) be a linear mapping.Then T is unital, surjective and spectrally bounded if and only if there area Jordan automorphism S of Mn(C) and a non-zero complex number γ suchthatTx = γ Sx+ (1−γ) τ(x) , x ∈ Mn(C) ,where τ denotes the normalised centre-valued trace on Mn(C).Corollary 1. The conjecture holds for T : A −→ Mn with A aC∗-algebra.Corollary 2. ([16]) The conjecture holds for T : A −→ B withdim(B) < ∞.Question 10. Does the above proposition hold if Mn(C) is replaced by avon Neumann factor of type II1?A positive answer would imply that the above conjecture holds for all finitevon Neumann factors.5. Existence of a universal class of m-isometriesVladiḿır Müller (Czech Academy of Sciences)A bounded linear operator T acting on a Hilbert space X is called anm-isometry if it satisfies the following identity:m∑k=0(−1)m−k(mk)T ∗kT k = 0 ,where T ∗ denotes the adjoint operator of T .We refer to [1, 2, 3, 4] for the study of m-isometries on Hilbert spaces.open problems in operator theory 155Question 11. Is there a universal operator for all m-isometries? Equiv-alently, is there an m-isometry A ∈ L(K) such that for any m-isometry T ona separable Hilbert space H there exist a subspace M ⊂ K invariant for Asuch that T is unitarily equivalent to A|M?6. Operators on ℓ∞ with dense rangeAmir-Bahman Nasseri (Bergische Universität Wuppertal)Tauberian operators were introduced in [14] as those bounded operatorsT : X → Y between Banach spaces X and Y such that the second conjugatesatisfies T ∗∗−1(Y ) = X.The following question is related with that of the existence of tauberianoperators T : L1(0, 1) → L1(0, 1) with non-closed range.Question 12. Let T : L1(0, 1) → L1(0, 1) be a bounded operator. Sup-pose that for all the subspaces M of L1(0, 1) that are isometric to ℓ1(N),the restriction T |M is an isomorphism (i.e., bounded below). Is T itself anisomorphism?It is possible to show that the previous question admits an equivalentformulation:Question 13. Let T : ℓ∞(N) → ℓ∞(N) be a bounded operator with denserange. Is T surjective?The latter formulation of the question was posed in MathOverflow [17].Thanks are due to W.B. Johnson for some useful comments.References[1] J. Agler, A disconjugacy theorem for Toeplitz operators, Amer. J. Math.112 (1) (1990), 1 – 14.[2] J. Agler, M. Stankus, m-isometric transformations of Hilbert space I,Integral Equations Operator Theory 21 (4) (1995), 383 – 429.[3] J. Agler, M. Stankus, m-isometric transformations of Hilbert space II,Integral Equations Operator Theory 23 (1) (1995), 1 – 48.[4] J. Agler, M. Stankus, m-isometric transformations of Hilbert space III,Integral Equations Operator Theory 24 (4) (1996), 379 – 421.[5] P. Aiena, “Fredholm and Local Spectral Theory, with Applications toMultipliers ”, Kluwer Academic Publishers, Dordrecht, 2004.156 t. bermúdez, m. gonzález, j. pello[6] J. Bès, J.A. Conejero, Hypercyclic subspaces in omega, J. Math. Anal.Appl. 316 (1) (2006), 16 – 23.[7] J. Bonet, A problem on the structure of Fréchet spaces, Rev. R. Acad. Cienc.Exactas F́ıs. Nat. Ser. A Math. RACSAM 104 (2010), 427 – 434.[8] J. Bonet, F. Mart́ınez-Giménez, A. Peris, A Banach space whichadmits no chaotic operator, Bull. London Math. Soc. 33 (2001), 196 – 198.[9] M. de la Rosa, L. Frerick, S. Grivaux, A. Peris, Frequent hyper-cyclicity, chaos, and unconditional Schauder decompositions, Israel J. Math.190 (2012), 389 – 399.[10] L. Frerick, A. Peris, personal communication, 2012.[11] G. Godefroy, J.H. Shapiro, Operators with dense, invariant, cyclic vectormanifolds, J. Funct. Anal. 98 (1991), 229 – 269.[12] K. Goebel, “Concise Course on Fixed Point Theorems ”, Yokohama Publish-ers, Yokohama, 2002.[13] K.-G. Grosse-Erdmann, A. Peris Manguillot, “Linear Chaos ”,Universitext Springer, London, 2011.[14] N. Kalton, A. Wilansky, Tauberian operators on Banach spaces, Proc.Amer. Math. Soc. 57 (1976), 251 – 255.[15] M. Mathieu, A collection of problems on spectrally bounded operators,Asian-Eur. J. Math. 2 (3) (2009), 487 – 501.[16] M. Mathieu, A.R. Sourour, Hereditary properties of spectral isometries,Arch. Math. (Basel) 82 (3) (2004), 222 – 229.[17] http://mathoverflow.net/questions/101253/surjectivity-of-operators-on-linfty

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Several open problems in operator theory

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