Norm-attaining compact operators

We show examples of compact linear operators between Banach spaces which cannot be approximated by norm attaining operators. This is the negative answer to an open question posed in the 1970's. Actually, any strictly convex Banach space failing the approximation property serves as the range space. On the other hand, there are examples in which the domain space has Schauder basis. It now makes sense to discuss sufficient conditions on the domain or the range space to ensure that every compact linear operator between them can be approximated by norm attaining operators. We get several basic results in this line and mention some open problems.Supported by Spanish MICINN and FEDER project no. MTM2012-31755 and by Junta de Andaluc´ıa and FEDER grants FQM-185 and P09-FQM-4911

Daugavet property in projective symmetric tensor products of Banach spaces

Miguel Martín partially supported by Spanish AEI Project PGC2018-093794-B- I00/AEI/10.13039/501100011033 (MCIU/AEI/FEDER, UE), A-FQM-484-UGR18 (Universidad de Granada and Junta de Analucía/FEDER, UE), FQM-185 (Junta de Andalucía/FEDER, UE), and by “Maria de Maeztu” Excellence Unit IMAG, reference CEX2020-001105-M funded by MCIN/AEI/10.13039/501100011033. Abraham Rueda Zoca was supported by Juan de la Cierva-Formación fellowship FJC2019-039973, by MTM2017-86182-P (Government of Spain, AEI/FEDER, EU), by Spanish AEI Project PGC2018-093794-B- I00/AEI/10.13039/501100011033 (MCIU/AEI/FEDER, UE), by Fundación Séneca, ACyT Región de Murcia grant 20797/PI/18, by Junta de Andalucía Grant A-FQM-484-UGR18 and by Junta de Andalucía Grant FQM-0185.We show that all the symmetric projective tensor products of a Banach space X have the Daugavet property provided X has the Daugavet property and either X is an L1-predual (i.e., X∗ is isometric to an L1-space) or X is a vector-valued L1-space. In the process of proving it, we get a number of results of independent interest. For instance, we characterise “localised” versions of the Daugavet property [i.e., Daugavet points and Δ-points introduced in Abrahamsen et al. (Proc Edinb Math Soc 63:475–496 2020)] for L1-preduals in terms of the extreme points of the topological dual, a result which allows to characterise a polyhedrality property of real L1-preduals in terms of the absence of Δ-points and also to provide new examples of L1-preduals having the convex diametral local diameter two property. These results are also applied to nicely embedded Banach spaces [in the sense of Werner (J Funct Anal 143:117–128, 1997)] so, in particular, to function algebras. Next, we show that the Daugavet property and the polynomial Daugavet property are equivalent for L1-preduals and for spaces of Lipschitz functions. Finally, an improvement of recent results in Rueda Zoca (J Inst Math Jussieu 20(4):1409–1428, 2021) about the Daugavet property for projective tensor products is also obtained.ACyT Región de Murcia 20797/PI/18Universidad de Granada and Junta de AnalucíaFundación SénecaEuropean CommissionEuropean Regional Development FundJunta de Andalucía FJC2019-039973, FQM-0185, MCIN/AEI/10.13039/501100011033, MTM2017-86182-PUniversity of the East FQM-18

Some stability properties for the Bishop-Phelps-Bollobás property for Lipschitz maps

The authors would like to thank A. Aviles, L. C. Garcia-Lirola, V. Kadets, P. Koszmider, and A. Rueda Zoca for kindly answering some inquiries about the contents of the paper. They also thank G. Choi, Y. S. Choi, and M. Jung for some comments. Finally, they thank the anonymous referee for the careful checking of the manuscript and multiple suggestions which have improved the final form of the paper. This research has been partially supported by Spanish AEI Project PGC2018-093794-B-I00/AEI/10.13039/501100011033 (MCIU/AEI/FEDER, UE) and by Junta de Andalucia/FEDER, UE project FQM-185.We study the stability behavior of the Bishop-Phelps-Bollobas property for Lipschitz maps (Lip-BPB property). This property is a Lipschitz version of the classical Bishop-Phelps-Bollobas property and deals with the possibility of approximating a Lipschitz map that almost attains its (Lipschitz) norm at a pair of distinct points by a Lipschitz map attaining its norm at a pair of distinct points (relatively) very close to the previous one. We first study the stability of this property under the (metric) sum of the domain spaces. Next, we study when it is possible to pass the Lip-BPB property from scalar functions to some vector-valued maps, getting some positive results related to the notions of Gamma-flat operators and ACK structure. We get sharper results for the case of Lipschitz compact maps. The behavior of the property with respect to absolute sums of the target space is also studied. We also get results similar to the above for the density of strongly norm attaining Lipschitz maps and of Lipschitz compact maps.Spanish AEI Project PGC2018-093794-B-I00/AEI/10.13039/501100011033Junta de Andalucia/FEDER , UE project FQM-18

The Daugavet equation for polynomials on C⁎-algebras and JB⁎-triples

We prove that every JB∗-triple E (in particular, every C∗- algebra) satisfying the Daugavet property also satisfies the stronger polynomial Daugavet property, that is, every weakly compact polynomial P : E −→ E satisfies the Daugavet equation IdX +P = 1 + P . The analogous conclusion also holds for the alternative Daugavet propert

Residuality in the set of norm attaining operators between Banach spaces

This paper was partially written when the first author was visiting the University of Granada and he would like to acknowledge the hospitality that he received there. The authors would like to thank Antonio Avilés, Luis Carlos García-Lirola, Gilles Godefroy, Manuel Maestre, Warren Moors, Vicente Montesinos, and Rafael Payá for kindly answering several inquiries related to the topics of the paper. We also thank the anonymous referee for the careful reading of the manuscript and for providing a number of comments which have improved its final form. M. Jung was supported by NRF (NRF-2019R1A2C1003857), by POSTECH Basic Science Research Institute Grant (NRF-2021R1A6A1A10042944) and by a KIAS Individual Grant (MG086601) at Korea Institute for Advanced Study. M. Martín was supported by Project PGC2018-093794-B-I00/AEI/10.13039/501100011033 (MCIU/AEI/FEDER, UE), by Junta de Andalucía I+D+i grants P20_00255, A-FQM-484-UGR18, and FQM-185, and by “Maria de Maeztu” Excellence Unit IMAG, reference CEX2020-001105-M funded by MCIN/AEI/10.13039/501100011033. A. Rueda Zoca was supported by Projects MTM2017-86182-P (Government of Spain, AEI/FEDER, EU), PGC2018-093794-B-I00/AEI/10.13039/501100011033 (MCIU/AEI/FEDER, UE), by Fundación Séneca, ACyT Región de Murcia grant 20797/PI/18, by Junta de Andalucía Grant A-FQM-484-UGR18, and by Junta de Andalucía Grant FQM-0185.We study the relationship between the residuality of the set of norm attaining functionals on a Banach space and the residuality and the denseness of the set of norm attaining operators between Banach spaces. Our first main result says that if C is a bounded subset of a Banach space X which admit an LUR renorming satisfying that, for every Banach space Y, the operators T from X to Y for which the supremum of with is attained are dense, then the set of those functionals which strongly exposes C is dense in ⁎. This extends previous results by J. Bourgain and K.-S. Lau. The particular case in which C is the unit ball of X, in which we get that the norm of ⁎ is Fréchet differentiable at a dense subset, improves a result by J. Lindenstrauss and we even present an example showing that Lindenstrauss' result was not optimal. In the reverse direction, we obtain results for the density of the set of absolutely strongly exposing operators from X to Y by requiring that the set of strongly exposing functionals on X is dense and conditions on Y or ⁎ involving RNP and discreteness on the set of strongly exposed points of Y or ⁎. These results include examples in which even the denseness of norm attaining operators was unknown. We also show that the residuality of the set of norm attaining operators implies the denseness of the set of absolutely strongly exposing operators provided the domain space and the dual of the range space are separable, extending a recent result for functionals. Finally, our results find important applications to the classical theory of norm-attaining operators, to the theory of norm-attaining bilinear forms, to the geometry of the preduals of spaces of Lipschitz functions, and to the theory of strongly norm-attaining Lipschitz maps. In particular, we solve a proposed open problem showing that the unique predual of the space of Lipschitz functions from the Euclidean unit circle fails to have Lindenstrauss property A.ACyT Región de Murcia 20797/PI/18Junta de Andalucía I+D+i A-FQM-484-UGR18, FQM-185, MCIN/AEI/10.13039/501100011033, MTM2017-86182-P, P20_00255KIAS MG086601POSTECH Basic Science Research Institute NRF-2021R1A6A1A10042944Institute for Advanced Study PGC2018-093794-B-I00/AEI/10.13039/501100011033Fundación SénecaEuropean CommissionFEDERJunta de Andalucía FQM-018

Diametral notions for elements of the unit ball of a Banach space

The first and third authors were supported by grant PID2021-122126NB-C31 funded by MICIU/AEI/10.13039/501100011033 and by ERDF/EU, by Junta de Andalucía I+D+i grants P20_00255 and FQM-185, and by “Maria de Maeztu” Excellence Unit IMAG (CEX2020-001105-M) funded by MICIU/AEI/10.13039/501100011033. The second named author was supported by the Estonian Research Council grant SJD58.We introduce extensions of Δ-points and Daugavet points in which slices are replaced by relatively weakly open subsets (super Δ-points and super Daugavet points) or by convex combinations of slices (ccs Δ-points and ccs Daugavet points). These notions represent the extreme opposite to denting points, points of continuity, and strongly regular points. We first give a general overview of these new concepts and provide some isometric consequences on the spaces. As examples: (1) If a Banach space contains a super Δ-point, then it does not admit an unconditional FDD (in particular, unconditional basis) with suppression constant smaller than 2. (2) If a real Banach space contains a ccs Δ-point, then it does not admit a one-unconditional basis. (3) If a Banach space contains a ccs Daugavet point, then every convex combination of slices of its unit ball has diameter 2. We next characterize the notions in some classes of Banach spaces, showing, for instance, that all the notions coincide in L1-predual spaces and that all the notions but ccs Daugavet points coincide in L1-spaces. We next comment on some examples which have previously appeared in the literature, and we provide some new intriguing examples: examples of super Δ-points which are as close as desired to strongly exposed points (hence failing to be Daugavet points in an extreme way); an example of a super Δ-point which is strongly regular (hence failing to be a ccs Δ-point in the strongest way); a super Daugavet point which fails to be a ccs Δ-point. The extensions of the diametral notions to points in the open unit ball and consequences on the spaces are also studied. Lastly, we investigate the Kuratowski measure of relatively weakly open subsets and of convex combinations of slices in the presence of super Δ-points or ccs Δ-points, as well as for spaces enjoying diameter-two properties. We conclude the paper with some open problems.MICIU/AEI/10.13039/501100011033 PID2021-122126NB-C31ERDF/EUJunta de Andalucía I+D+i P20_00255, FQM-185MICIU/AEI/10.13039/501100011033 “Maria de Maeztu” (CEX2020-001105-M

Dynamics, Operator Theory, and Infinite Holomorphy

The works on linear dynamics in the last two decades show that many, even quite natural, linear dynamical systems exhibit wild behaviour. Linear chaos and hypercyclicity have been at the crossroads of several areas of mathematics. More recently, fascinating new connections have started to be explored: operators on spaces of analytic functions, semigroups and applications to partial differential equations, complex dynamics, and ergodic theory. Related aspects of functional analysis are the study of linear operators on Banach spaces by using geometric, topological, and algebraic techniques, the works on the geometry of Banach spaces and Banach algebras, and the study of the geometry of a Banach space via the behaviour of some of its operators. In recent years some aspects of the theory of infinite-dimensional complex analysis have attracted the attention of several researchers. One is in the general field of Banach and Fréchet algebras and Banach spaces of polynomial and holomorphic functions. Another is in a deep connection with the theory of one and several complex variables as Dirichlet series in one variable, Bohr radii in several variables, Bohnenblust-Hille constants, Sidon constants, domains of convergence, and so forth. This special issue shows some new advances in the topics shortly described above

Characterization of the chemical structure of vinyl ester resin in a climate chamber under different conditions of degradation

Due to the good strength and similar toughness of epoxy resins, vinyl ester resins are widely used as thermoset adhesives in structural adhesive joints and as composites for different industrial applications. However, vinyl ester adhesives are difficult to cure completely under environmental conditions, even after long periods of time because of gel formation slows the necessary diffusion of the catalyst across the polymer network. Several studies have used weathering chambers to investigate the degradation mechanisms of vinyl ester adhesives. However, a review of the scientific literature revealed both a wide variety of aging processes and several ambiguities between the recorded experimental results. In this work, post-cured vinyl ester resins at different aging cycles were aged under high temperature and relative humidity, and the changes in their structure, mechanical and adhesion properties were studied. Chemical and structural changes were observed in the vinyl ester resins after aging in a climatic chamber

On the numerical index with respect to an operator

The research of the first author is done in frames of Ukrainian Ministry of Science and Education Research Pro gram 0118U002036, and it was partially done during his stay in the University of Granada which was supported by the project MTM2015-65020-P (MINECO/FEDER, UE). Research of second, third, and fifth authors is supported by projects MTM2015-65020-P (MINECO/FEDER, UE), PGC2018-093794-B-I00 (MCIU/AEI/FEDER, UE), and FQM-185 (Junta de Andalucía/FEDER, UE). The fourth author acknowledges financial support from the Spanish Ministry of Economy and Com petitiveness, through the “Severo Ochoa Programme for Centres of Excellence in R&D” (SEV-2015-0554).The aim of this paper is to study the numerical index with respect to an operator between Banach spaces. Given Banach spaces X and Y, and a norm-one operator G∈L(X,Y) (the space of all bounded linear operators from X to Y), the numerical index with respect to G, nG(X,Y), is the greatest constant k≥0 such that k∥T∥≤infδ>0sup{|y∗(Tx)|:y∗∈Y∗,x∈X,∥y∗∥=∥x∥=1,Rey∗(Gx)>1−δ} for every T∈L(X,Y). Equivalently, nG(X,Y) is the greatest constant k≥0 such that max∣∣w∣∣=1∥G+wT∥≥1+k∥T∥ for all T∈L(X,Y). Here, we first provide some tools to study the numerical index with respect to G. Next, we present some results on the set N(L(X,Y)) of the values of the numerical indices with respect to all norm-one operators in L(X,Y). For instance, N(L(X,Y))={0} when X or Y is a real Hilbert space of dimension greater than 1 and also when X or Y is the space of bounded or compact operators on an infinite-dimensional real Hilbert space. In the real case N(L(X,ℓp))⊆[0,Mp]andN(L(ℓp,Y))⊆[0,Mp] for 1<p<∞ and for all real Banach spaces X and Y, where Mp=supt∈[0,1]∣∣tp−1−t∣∣1+tp. For complex Hilbert spaces H1, H2 of dimension greater than 1, N(L(H1,H2))⊆{0,1/2} and the value 1/2 is taken if and only if H1 and H2 are isometrically isomorphic. Moreover, N(L(X,H))⊆[0,1/2] and N(L(H,Y))⊆[0,1/2] when H is a complex infinite-dimensional Hilbert space and X and Y are arbitrary complex Banach spaces. Also, N(L(L1(μ1),L1(μ2)))⊆{0,1} and N(L(L∞(μ1),L∞(μ2)))⊆{0,1} for arbitrary σ-finite measures μ1 and μ2, in both the real and the complex cases. Also, we show that the Lipschitz numerical range of Lipschitz maps from a Banach space to itself can be viewed as the numerical range of convenient bounded linear operators with respect to a bounded linear operator. Further, we provide some results which show the behaviour of the value of the numerical index when we apply some Banach space operations, such as constructing diagonal operators between c0-, ℓ1-, or ℓ∞-sums of Banach spaces, composition operators on some vector-valued function spaces, taking the adjoint to an operator, and composition of operators.Ukrainian Ministry of Science and Education Research Program 0118U002036MINECO/FEDER, UE MTM2015-65020-PJunta de Andalucia/FEDER, UE FQM-185MCIU/AEI/FEDER, UE PGC2018-093794-B-I00Spanish Ministry of Economy and Competitiveness, through the "Severo Ochoa Programme for Centres of Excellence in RD" SEV-2015-055

Luxación traumática de la rodilla

La luxación traumática de la rodilla es la lesión poco frecuente que se asocia a grandes lesiones cápsuloligamentosas y que a menudo afecta a estructuras neurológicas y vasculares. La lesión vascular es la complicación más importante. Todos los autores coinciden en que el diagnóstico precoz y tratamiento urgente de las lesiones de la arteria poplítea son esenciales. Sin embargo, no existe el mismo acuerdo en cuanto al tratamiento de las lesiones cápsuloligamentosas. Algunos autores publican buenos resultados con tratamientos no quirúrgicos, aunque otros consideran que el mejor tratamiento es la reparación completa de las lesiones ligamentosas. Hemos tratado en nuestro hospital siete casos de luxación traumática de rodilla en los últimos 20 años. Presentamos el tratamiento realizado, sus complicaciones y resultados.Traumatic knee dislocation is an uncommon injury associated with extensive soft tissue damage and often neurovascular involvement. Vascular injury is the most serious complication. There is genera agreement that early diagnosis and immediate treatment of popliteal artery disruption is essential. However there is no consensus about treatment of capsular and ligamentous injuries. Some authors reported good results advocated the repair of all ligaments in the best method. Seven traumatic knee dislocations have been treated in our hospital for last 20 years. Initial treatment, complications and results are presented