Absolutely (q, 1)-summing operators acting in C(K)-spaces and the weighted Orlicz property for Banach spaces

[EN] We provide a new separation-based proof of the domination theorem for (q, 1)-summing operators. This result gives the celebrated factorization theorem of Pisier for (q, 1)-summing operators acting in C(K)-spaces. As far as we know, none of the known versions of the proof uses the separation argument presented here, which is essentially the same that proves Pietsch Domination Theorem for p-summing operators. Based on this proof, we propose an equivalent formulation of the main summability properties for operators, which allows to consider a broad class of summability properties in Banach spaces. As a consequence, we are able to show new versions of the Dvoretzky-Rogers Theorem involving other notions of summability, and analyze some weighted extensions of the q-Orlicz property.Both authors were supported by the Ministerio de Ciencia, Innovacion y Universidades, Agencia Estatal de Investigacion (Spain) and FEDER, the first author under project PGC2018-095366-B-100 and the second under project MTM2016-77054-C2-1-P.Calabuig, JM.; Sánchez Pérez, EA. (2021). Absolutely (q, 1)-summing operators acting in C(K)-spaces and the weighted Orlicz property for Banach spaces. Positivity. 25(3):1199-1214. https://doi.org/10.1007/s11117-021-00811-yS1199121425

Final outcome of raptors admitted to the Tafira Wildlife Rehabilitation Center, Gran Canaria Island, Spain (2003–2013)

Disposición final de las rapaces admitidas en el Centro de Rehabilitación de Fauna Silvestre de Tafira, Gran Canaria, España (2003–2013) En este estudio se analizan los resultados de la rehabilitación de aves rapaces silvestres admitidas en el Centro de Rehabilitación de Fauna Silvestre de Tafira en la isla de Gran Canaria, España, desde 2003 hasta 2013, mediante un sistema de auditoría de la calidad basado en los índices generales y clasificados (por causa de admisión) de las cuatro categorías de disposición final, el tiempo hasta la muerte y el tiempo de permanencia como indicadores de calidad. Las categorías de disposición final fueron: eutanasia (Er) = 19,78%, mortalidad no asistida durante la hospitalización (Mr) = 22,20%, liberación (Rr) = 57,57% y cautividad permanente (Cr) = 0,46%. Teniendo en cuenta la especial vulnerabilidad de las especies de rapaces insulares y el alto índice de Rr conseguido, los resultados de este estudio ponen de relieve la importancia de los centros de rehabilitación de fauna silvestre en el tratamiento médico de las aves heridas y la consiguiente liberación de las aves rehabilitadas.Disposición final de las rapaces admitidas en el Centro de Rehabilitación de Fauna Silvestre de Tafira, Gran Canaria, España (2003–2013) En este estudio se analizan los resultados de la rehabilitación de aves rapaces silvestres admitidas en el Centro de Rehabilitación de Fauna Silvestre de Tafira en la isla de Gran Canaria, España, desde 2003 hasta 2013, mediante un sistema de auditoría de la calidad basado en los índices generales y clasificados (por causa de admisión) de las cuatro categorías de disposición final, el tiempo hasta la muerte y el tiempo de permanencia como indicadores de calidad. Las categorías de disposición final fueron: eutanasia (Er) = 19,78%, mortalidad no asistida durante la hospitalización (Mr) = 22,20%, liberación (Rr) = 57,57% y cautividad permanente (Cr) = 0,46%. Teniendo en cuenta la especial vulnerabilidad de las especies de rapaces insulares y el alto índice de Rr conseguido, los resultados de este estudio ponen de relieve la importancia de los centros de rehabilitación de fauna silvestre en el tratamiento médico de las aves heridas y la consiguiente liberación de las aves rehabilitadas.The outcomes of wild raptors admitted to the Tafira Wildlife Rehabilitation Center in Gran Canaria Island, Spain, from 2003 to 2013 were analyzed using a quality auditing system based on the crude and stratified (by causes of admission) rates of four final outcome categories, time until death and length of stay as quality indicators. The outcome categories were: euthanasia (Er = 19.78%); unassisted mortality during hospitalization (Mr = 22.20%); release (Rr = 57.57%); and permanent captivity (Cr = 0.46%). Taking into account the particular vulnerability of insular raptor species and the high Rr achieved, findings from this study emphasize the importance of wildlife rehabilitation centers for the medical management of injured raptors and the subsequent release of rehabilitated individuals into the wild

Extending and factorizing bounded bilinear maps defined on order continuous Banach function spaces

We consider the problem of extending or factorizing a bounded bilinear map defined on a couple of order continuous Banach function spaces to its optimal domain, i.e. the biggest couple of Banach function spaces to which the bilinear map can be extended. As in the case of linear operators, we use vector measure techniques to find this space, and we show that this procedure cannot be always successfully used for bilinear maps. We also present some applications to find optimal factorizations of linear operators between Banach function spaces.J. M. Calabuig was supported by Ministerio de Economia y Competitividad (Spain) (project MTM2011-23164) and by "Jose Castillejo 2009" (MEC). E. A. Sanchez-Perez was supported by MEC and FEDER (project MTM2009-14483-C02-02). J. M. Calabuig and E. A. Sanchez-Perez were also supported by Ayuda para Estancias de PDI de la UPV en Centros de Investigacion de Prestigio (PAID-00-11).Calabuig Rodriguez, JM.; Fernandez Unzueta, M.; Galaz Fontes, F.; Sánchez Pérez, EA. (2014). Extending and factorizing bounded bilinear maps defined on order continuous Banach function spaces. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas (RACSAM). 108(2):353-367. https://doi.org/10.1007/s13398-012-0101-7S3533671082Calabuig, J.M., Galaz-Fontes, F., Jiménez Fernández, E., Sánchez Pérez, E.A.: Strong factorization of operators on spaces of vector measure integrable functions and unconditional convergence of series. Math. Z. 257, 381–402 (2007)Calabuig, J.M., Delgado, O., Sánchez Pérez, E.A.: Factorizing operators on Banach function spaces through spaces of multiplication operators. J. Math. Anal. Appl. 364, 88–103 (2010)Curbera, G.P.: Operators into $$L^1$$ of a vector measure and applications to Banach lattices. Math. Ann. 293, 317–330 (1992)Curbera, G.P., Ricker, W.J.: Optimal domains for kernel operators via interpolation. Math. Nachr. 244, 47–63 (2002)Curbera, G.P. , Ricker, W.J.: Optimal domains for the kernel operator associated with Sobolev’s inequality. Studia Math. 158(2), 131–152 (2003) [see also Corrigenda in the same journal, 170 (2005) 217–218)]Delgado, O.: Banach function subspaces of $$L^1$$ of a vector measure and related Orlicz spaces. Indag. Math. (N. S.) 15, 485–495 (2004)Delgado, O.: Optimal domains for kernel operators on $$[0,\infty )\times [0,\infty )$$ . Studia Math. 174, 131–145 (2006)Delgado, O., Soria, J.: Optimal domain for the Hardy operator. J. Funct. Anal. 244, 119–133 (2007)Diestel, J., Uhl, J.J.: Vector measures. In: Math. Surveys, vol. 15. Amer. Math. Soc., Providence (1977)Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces II. Springer, Berlin (1979)Galdames, O., Sánchez Pérez, E.A.: Optimal range theorems for operators with $$p$$ -th power factorable adjoints. Banach J. Math. Anal. 6(1), 61–73 (2012)Okada, S., Ricker, W.J., Sánchez Pérez, E.A.: Optimal domain and integral extension of operators acting in function spaces. In: Oper. Theory Adv. Math. Appl., vol. 180. Birkäuser, Basel (2008

On p-Dunford integrable functions with values in Banach spaces

[EN] Let (Omega, Sigma, mu) be a complete probability space, X a Banach space and 1 X. Special attention is paid to the compactness of the Dunford operator of f. We also study the p-Bochner integrability of the composition u o f: Omega->Y , where u is a p-summing operator from X to another Banach space Y . Finally, we also provide some tests of p-Dunford integrability by using w*-thick subsets of X¿.Research partially supported by Ministerio de Economia, Industria y Competitividad and FEDER under projects MTM2014-53009-P (J.M. Calabuig), MTM2014-54182-P (J. Rodriguez) and MTM2016-77054-C2-1-P (P. Rueda and E.A. Sanchez-Perez). The second author was also partially supported by project 19275/PI/14 funded by Fundacion Seneca - Agencia de Ciencia y Tecnologia de la Region de Murcia within the framework of PCTIRM 2011-2014.Calabuig, JM.; Rodríguez, J.; Rueda, P.; Sánchez Pérez, EA. (2018). On p-Dunford integrable functions with values in Banach spaces. Journal of Mathematical Analysis and Applications. 464(1):806-822. https://doi.org/10.1016/j.jmaa.2018.04.030S806822464

Tensor product representation of Kothe-Bochner spaces and their dual spaces

We provide a tensor product representation of Kothe-Bochner function spaces of vector valued integrable functions. As an application, we show that the dual space of a Kothe-Bochner function space can be understood as a space of operators satisfying a certain extension property. We apply our results in order to give an alternate representation of the dual of the Bochner spaces of p-integrable functions and to analyze some properties of the natural norms that are defined on the associated tensor products.First and third authors are supported by grant MTM201453009-P of the Ministerio de Economia y Competitividad (Spain). Second and fourth authors are supported by grant MTM2012-36740-C02-02 of the Ministerio de Economia y Competitividad (Spain).Calabuig, JM.; Jiménez Fernández, E.; Juan Blanco, MA.; Sánchez Pérez, EA. (2016). Tensor product representation of Kothe-Bochner spaces and their dual spaces. Positivity. 20(1):155-169. https://doi.org/10.1007/s11117-015-0347-3S155169201Bochner, S.: Integration von Funktionen, deren Werte die Elemente eines Vectorraumes sind. Fundamenta Mathematicae 20, 262–276 (1933)Calabuig, J.M., Delgado, O., Juan, M.A., Sánchez, E.A.: Pérez, On the Banach lattice structure of $$L^1_w$$ L w 1 of a vector measure on a $$\delta$$ δ -ring. Collect. Math. 65, 6567–85 (2014)Calabuig, J.M., Delgado, O., Sánchez Pérez, E.A.: Factorizing operators on Banach function spaces through spaces of multiplication operators. J. Math. Anal. Appl. 364(1), 88–103 (2010)Calabuig, J.M., Gregori, P., Sánchez, E.A.: Pérez, Radon-Nikodým derivatives for vector measures belonging to Köthe function spaces. J. Math. Anal. Appl. 348, 469–479 (2008)Cerdà, J., Hudzik, H., Mastyło, M.: Geometric properties of Köthe-Bochner spaces. Math. Proc. Cambridge Philos. Soc. 120(3), 521–533 (1996)Chakraborty, N.D., Basu, S.: Spaces of p-tensor integrable functions and related Banach space properties. Real Anal. Exchange 34, 87–104 (2008)Chakraborty, N.D., Basu, S.: Integration of vector-valued functions with respect to vector measures defined on $$\delta$$ δ -rings. Ill. J. Math. 55(2), 495–508 (2011)Defant, A., Floret, K.: Tensor norms and operator ideals. North-Holland, Amsterdam (1993)Delgado, O., Juan, M.A.: Representation of Banach lattices as $$L^{1}_{w}$$ L w 1 spaces of a vector measure defined on a $$\delta -$$ δ - ring. Bull. Belgian Math. Soc. 19, 239–256 (2012)Diestel, J., Uhl, J.J.: Vector measures. Am. Math. Soc, Providence (1977)Dobrakov, I.: On integration in Banach spaces, VII. Czechoslovak Math. J. 38, 434–449 (1988)García-Raffi, L.M., Jefferies, B.: An application of bilinear integration to quantum scattering. J. Math. Anal. Appl. 415, 394–421 (2014)Gregori Huerta, P.: Espacios de medidas vectoriales. Thesis, Universidad de Valencia, ISBN:8437060591 (2005)Jefferies, B., Okada, S.: Bilinear integration in tensor products. Rocky Mt. J. Math. 28, 517–545 (1998)Lewis, D.R.: On integrability and summability in vector spaces. Ill. J. Math. 16, 294–307 (1972)Lin, P.-K.: Köthe-Bochner function spaces. Birkhauser, Boston (2004)Lindenstrauss, J., Tzafriri, L.: Classical Banach spaces II. Springer, Berlin (1979)Okada, S., Ricker, W.J., Sánchez Pérez, E.A.: Optimal Domains and integral extensions of operators acting in function spaces. Operator Theory Advances and Applications, vol. 180. Birkhäuser, Basel (2008)Pallu de La Barriére, R.: Integration of vector functions with respect to vector measures. Studia Univ. Babes-Bolyai Math. 43, 55–93 (1998)Rodríguez, J.: On integration of vector functions with respect to vector measures. Czechoslovak Math. J. 56, 805–825 (2006

Factorizing operators on Banach function spaces through spaces of multiplication operators

In order to extend the theory of optimal domains for continuous operators on a Banach function space X(μ) over a finite measure μ, we consider operators T satisfying other type of inequalities than the one given by the continuity which occur in several well-known factorization theorems (for instance, Pisier Factorization Theorem through Lorentz spaces, pth-power factorable operators . . . ). We prove that such a T factorizes through a space of multiplication operators which can be understood in a certain sense as the optimal domain for T . Our extended optimal domain technique does not need necessarily the equivalence between μ and the measure defined by the operator T and, by using δ-rings, μ is allowed to be infinite. Classical and new examples and applications of our results are also given, including some new results on the Hardy operator and a factorization theorem through Hilbert spaces.Generalitat Valenciana TSGD-07Ministerio de Educación y Ciencia MTM2006-13000-C03-0

Vector-valued impact measures and generation of specific indexes for research assessment

A mathematical structure for defining multi-valued bibliometric indices is provided with the aim of measuring the impact of general sources of information others than articles and journals-for example, repositories of datasets. The aim of the model is to use several scalar indices at the same time for giving a measure of the impact of a given source of information, that is, we construct vector valued indices. We use the properties of these vector valued indices in order to give a global answer to the problem of finding the optimal scalar index for measuring a particular aspect of the impact of an information source, depending on the criterion we want to fix for the evaluation of this impact. The main restrictions of our model are (1) it uses finite sets of scalar impact indices (altmetrics), and (2) these indices are assumed to be additive. The optimization procedure for finding the best tool for a fixed criterion is also presented. In particular, we show how to create an impact measure completely adapted to the policy of a specific research institution.Calabuig, JM.; Ferrer Sapena, A.; Sánchez Pérez, EA. (2016). Vector-valued impact measures and generation of specific indexes for research assessment. Scientometrics. 108(3):1425-1443. doi:10.1007/s11192-016-2039-6S142514431083Aleixandre Benavent, R., Valderrama Zurián, J. C., & González Alcaide, G. (2007). Scientific journals impact factor: Limitations and alternative indicators. El Profesional de la Información, 16(1), 4–11.Alguliyev, R., Aliguliyev, R. & Ismayilova, N. (2015). Weighted impact factor (WIF) for assessing the quality of scientific journals. arXiv:1506.02783Beauzamy, B. (1982). Introduction to Banach spaces and their geometry. Amsterdam: North-Holland.Beliakov, G., & James, S. (2011). Citation-based journal ranks: the use of fuzzy measures. Fuzzy Sets and Systems, 167, 101–119.Buela-Casal, G. (2003). Evaluating quality of articles and scientific journals. Proposal of weighted impact factor and a quality index. Psicothema, 15(1), 23–25.Diestel, J., & Uhl, J. J. (1977). Vector measures. Providence: Am. Math. Soc.Dorta-González, P., & Dorta-González, M. I. (2013). Comparing journals from different fields of science and social science through a JCR subject categories normalized impact factor. Scientometrics, 95(2), 645–672.Dorta-González, P., Dorta-González, M. I., Santos-Penate, D. R., & Suarez-Vega, R. (2014). Journal topic citation potential and between-field comparisons: The topic normalized impact factor. Journal of Informetrics, 8(2), 406–418.Egghe, L., & Rousseau, R. (2002). A general frame-work for relative impact indicators. Canadian Journal of Information and Library Science, 27(1), 29–48.Ferrer-Sapena, A., Sánchez-Pérez, E. A., González, L. M., Peset, F. & Aleixandre-Benavent, R. (2016). The impact factor as a measuring tool of the prestige of the journals in research assessment in mathematics. Research Evaluation, 1–9. doi: 10.1093/reseval/rvv041 .Ferrer-Sapena, A., Sánchez-Pérez, E. A., González, L. M., Peset, F., & Aleixandre-Benavent, R. (2015). Mathematical properties of weighted impact factors based on measures of prestige of the citing journals. Scientometrics, 105(3), 2089–2108.Gagolewski, M., & Mesiar, R. (2014). Monotone measures and universal integrals in a uniform framework for the scientific impact assessment problem. Information Sciences, 263, 166–174.Habibzadeh, F., & Yadollahie, M. (2008). Journal weighted impact factor: A proposal. Journal of Informetrics, 2(2), 164–172.Klement, E., Mesiar, R., & Pap, E. (2010). A universal integral as common frame for Choquet and Sugeno integral. IEEE Transactions on Fuzzy Systems, 18, 178–187.Leydesdorff, L., & Opthof, T. (2010). Scopus’s source normalized impact per paper (SNIP) versus a journal impact factor based on fractional counting of citations. Journal of the American Society for Information Science and Technology, 61, 2365–2369.Li, Y. R., Radicchi, F., Castellano, C., & Ruiz-Castillo, J. (2013). Quantitative evaluation of alternative field normalization procedures. Journal of Informetrics, 7(3), 746–755.Moed, H. F. (2010). Measuring contextual citation impact of scientific journals. Journal of Informetrics, 4, 265–277.Owlia, P., Vasei, M., Goliaei, B., & Nassiri, I. (2011). Normalized impact factor (NIF): An adjusted method for calculating the citation rate of biomedical journals. Journal of Biomedical Informatics, 44(2), 216–220.Pinski, G., & Narin, F. (1976). Citation influence for journal aggregates of scientific publications: Theory, with application to the literature of physics. Information Processing and Management, 12, 297–312.Piwowar, H. (2013). Altmetrics: Value all research products. Nature, 493(7431), 159–159.Pudovkin,A.I., & Garfield, E. (2004). Rank-normalized impact factor: A way to compare journal performance across subject categories. In Proceedings of the 67th annual meeting of the American Society for Information science and Technology, 41, 507-515.Rousseau, R. (2002). Journal evaluation: Technical and practical issues. Library Trends, 50(3), 418–439.Ruiz Castillo, J., & Waltman, L. (2015). Field-normalized citation impact indicators using algorithmically constructed classification systems of science. Journal of Informetrics, 9, 102–117.Torra, V., & Narukawa, Y. (2008). The h-index and the number of citations: Two fuzzy integrals. IEEE Transactions on Fuzzy Systems, 16, 795–797.Waltman, L., & van Eck, N. J. (2008). Some comments on the journal weighted impact factor proposed by Habibzadeh and Yadollahie. Journal of Informetrics, 2(4), 369–372.Waltman, L., & van Eck, N. J. (2010). The relation between Eigenfactor, audience factor, and influence weight. Journal of the American Society for Information Science and Technology, 61, 1476–1486.Zahedi, Z., Costas, R., & Wouters, P. (2014). How well developed are altmetrics? A cross-disciplinary analysis of the presence of ’alternative metrics’ in scientific publications. Scientometrics, 101(2), 1491–1513.Zitt, M. (2010). Citing-side normalization of journal impact: A robust variant of the Audience Factor. Journal of Informetrics, 4(3), 392–406.Zitt, M. (2011). Behind citing-side normalization of citations: Some properties of the journal impact factor. Scientometrics, 89, 329–344.Zitt, M., & Small, H. (2008). Modifying the journal impact factor by fractional citation weighting: The audience factor. Journal of the American Society for Information Science and Technology, 59, 1856–1860.Zyczkowski, K. (2010). Citation graph, weighted impact factors and performance indices. Scientometrics, 85(1), 301–315

Weighted p-regular kernels for reproducing kernel Hilbert spaces and Mercer Theorem

[EN] Let (X, Sigma, mu) be a finite measure space and consider a Banach function space Y(mu). Motivated by some previous papers and current applications, we provide a general framework for representing reproducing kernel Hilbert spaces as subsets of Kothe Bochner (vectorvalued) function spaces. We analyze operator-valued kernels Gamma that define integration maps L-Gamma between Kothe-Bochner spaces of Hilbert-valued functions Y(mu; kappa). We show a reduction procedure which allows to find a factorization of the corresponding kernel operator through weighted Bochner spaces L-P(gd mu; kappa) and L-P (hd mu; kappa) - where 1/p + 1/p' = 1 - under the assumption of p-concavity of Y(mu). Equivalently, a new kernel obtained by multiplying Gamma by scalar functions can be given in such a way that the kernel operator is defined from L-P (mu; kappa) to L-P (mu; kappa) in a natural way. As an application, we prove a new version of Mercer Theorem for matrix-valued weighted kernels.The second author acknowledges the support of the Ministerio de Economia y Competitividad (Spain), under project MTM2014-53009-P (Spain). The third author acknowledges the support of the Ministerio de Ciencia, Innovacion y Universidades (Spain), Agencia Estatal de Investigacion, and FEDER under project MTM2016-77054-C2-1-P (Spain).Agud Albesa, L.; Calabuig, JM.; Sánchez Pérez, EA. (2020). Weighted p-regular kernels for reproducing kernel Hilbert spaces and Mercer Theorem. Analysis and Applications. 18(3):359-383. https://doi.org/10.1142/S0219530519500179S359383183Agud, L., Calabuig, J. M., & Sánchez Pérez, E. A. (2011). The weak topology on q-convex Banach function spaces. Mathematische Nachrichten, 285(2-3), 136-149. doi:10.1002/mana.201000030CARMELI, C., DE VITO, E., & TOIGO, A. (2006). VECTOR VALUED REPRODUCING KERNEL HILBERT SPACES OF INTEGRABLE FUNCTIONS AND MERCER THEOREM. Analysis and Applications, 04(04), 377-408. doi:10.1142/s0219530506000838CARMELI, C., DE VITO, E., TOIGO, A., & UMANITÀ, V. (2010). VECTOR VALUED REPRODUCING KERNEL HILBERT SPACES AND UNIVERSALITY. Analysis and Applications, 08(01), 19-61. doi:10.1142/s0219530510001503Cerdà, J., Hudzik, H., & Mastyło, M. (1996). Geometric properties of Köthe–Bochner spaces. Mathematical Proceedings of the Cambridge Philosophical Society, 120(3), 521-533. doi:10.1017/s0305004100075058Chavan, S., Podder, S., & Trivedi, S. (2018). Commutants and reflexivity of multiplication tuples on vector-valued reproducing kernel Hilbert spaces. Journal of Mathematical Analysis and Applications, 466(2), 1337-1358. doi:10.1016/j.jmaa.2018.06.062Christmann, A., Dumpert, F., & Xiang, D.-H. (2016). On extension theorems and their connection to universal consistency in machine learning. Analysis and Applications, 14(06), 795-808. doi:10.1142/s0219530516400029Defant, A. (2001). Positivity, 5(2), 153-175. doi:10.1023/a:1011466509838Defant, A., & Sánchez Pérez, E. A. (2004). Maurey–Rosenthal factorization of positive operators and convexity. Journal of Mathematical Analysis and Applications, 297(2), 771-790. doi:10.1016/j.jmaa.2004.04.047De Vito, E., Umanità, V., & Villa, S. (2013). An extension of Mercer theorem to matrix-valued measurable kernels. Applied and Computational Harmonic Analysis, 34(3), 339-351. doi:10.1016/j.acha.2012.06.001Eigel, M., & Sturm, K. (2017). Reproducing kernel Hilbert spaces and variable metric algorithms in PDE-constrained shape optimization. Optimization Methods and Software, 33(2), 268-296. doi:10.1080/10556788.2017.1314471Fasshauer, G. E., Hickernell, F. J., & Ye, Q. (2015). Solving support vector machines in reproducing kernel Banach spaces with positive definite functions. Applied and Computational Harmonic Analysis, 38(1), 115-139. doi:10.1016/j.acha.2014.03.007Galdames Bravo, O. (2014). Generalized Kӧthe $p$-dual spaces. Bulletin of the Belgian Mathematical Society - Simon Stevin, 21(2). doi:10.36045/bbms/1400592625Lin, P.-K. (2004). Köthe-Bochner Function Spaces. doi:10.1007/978-0-8176-8188-3Lindenstrauss, J., & Tzafriri, L. (1979). Classical Banach Spaces II. doi:10.1007/978-3-662-35347-9Meyer-Nieberg, P. (1991). Banach Lattices. Universitext. doi:10.1007/978-3-642-76724-1Okada, S., Ricker, W. J., & Sánchez Pérez, E. A. (2008). Optimal Domain and Integral Extension of Operators. doi:10.1007/978-3-7643-8648-1Zhang, H., & Zhang, J. (2013). Vector-valued reproducing kernel Banach spaces with applications to multi-task learning. Journal of Complexity, 29(2), 195-215. doi:10.1016/j.jco.2012.09.00