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Lineability of the set of supporting vectors

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

The set of supporting vectors of a continuous linear operator \(T:X\rightarrow Y\) between normed spaces, denoted by \(\mathrm {suppv}(T)\) since 2017, is defined as \(\mathrm {suppv}(T):=\{x\in X:\Vert T(x)\Vert =\Vert T\Vert \Vert x\Vert \}\). In this manuscript, we study the lineability and coneability properties of \(\mathrm {suppv}(T)\), reaching both positive and negative results depending on T. For instance, if T is a functional, then \(\mathrm {suppv}(T)\) is 1-lineable and this result cannot be improved. However, if T is a (1, 1)-projection and X is infinite dimensional, then either \(\mathrm {suppv}(T)\) or \(\mathrm {suppv}(I-T)\) is lineable. Other general sufficient conditions for \(\mathrm {suppv}(T)\) to be lineable and coneable are provided. Special attention is given to the case where X and Y are Hilbert spaces, obtaining a characterization of unitary operators. The particular case of operators on \(\ell _1\) and \(c_0\) is also studied.

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References

  1. Acosta, M.D., Aizpuru, A., Aron, R.M., García-Pacheco, F.J.: Functionals that do not attain their norm. Bull. Belg. Math. Soc. Simon Stevin 14, 407–418 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aizpuru, A., García-Pacheco, F.J.: A short note about exposed points in real Banach spaces. Acta Math. Sci. Ser. B (Engl. Ed.) 28(4), 797–800 (2008)

    MathSciNet  MATH  Google Scholar 

  3. Aizpuru, A., Pérez-Eslava, C., García-Pacheco, F.J., Seoane-Sepúlveda, J.B.: Lineability and coneability of discontinuous functions on \(\mathbb{R}\). Publ. Math. Debrecen 72(1–2), 129–139 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Albkwre, G., Ciesielski, K., Wojciechowski, J.: Lineability of the functions that are Sierpiski–Zygmund, Darboux, but not connectivity. Rev. R. Acad. Cienc. Exactas Fs. Nat. Ser. A Mat. RACSAM 114(3), 145 (2020)

    Article  MATH  Google Scholar 

  5. Aron, R.M., Bernal-González, L., Pellegrino, D., Seoane-Sepúlveda, J.B.: Lineability: The Search for Linearity in Mathematics. Monographs and Research Notes in Mathematics. Chapman & Hall/CRC, Boca Raton (2015)

    Book  MATH  Google Scholar 

  6. Aron, R.M., García-Pacheco, F.J., Pérez-García, D., Seoane-Sepúlveda, J.B.: On dense-lineability of sets of functions on \(\mathbb{R}\). Topology 48(2–4), 149–156 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Aron, R.M., Gurariy, V.I., Seoane-Sepúlveda, J.B.: Lineability and spaceability of sets of functions on \(\mathbb{R}\). Proc. Am. Math. Soc. 133, 795–803 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Aron, R.M., Pérez-García, D., Seoane-Sepúlveda, J.B.: Algebrability of the set of non-convergent Fourier series. Studia Math. 175(1), 83–90 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Aron, R.M., Seoane-Sepúlveda, J.B.: Algebrability of the set of everywhere surjective functions on \(\mathbb{C}\). Bull. Belg. Math. Soc. Simon Stevin 14(1), 25–31 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bandyopadhyay, P., Godefroy, G.: Linear structure in the set of norm-attaining functionals on a Banach space. J. Convex Anal. 13, 489–497 (2006)

    MathSciNet  MATH  Google Scholar 

  11. Bartoszewicz, A., Gła̧bb, S., : Additivity and lineability in vector spaces. Linear Algebra Appl. 439(7), 2123–2130 (2013)

  12. Bernal-González, L., Pellegrino, D., Seoane-Sepúlveda, J.B.: Linear subsets of nonlinear sets in topological vector spaces. Bull. Am. Math. Soc. (N.S.) 51(1), 71–130 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bishop, E., Phelps, R.R.: A proof that every Banach space is subreflexive. Bull. Am. Math. Soc. 67, 97–98 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cobos Sánchez, C., García-Pacheco, F.J., Guerrero-Rodríguez, J.M., Hill, J.R.: An inverse boundary element method computational framework for designing optimal TMS coils. Eng. Anal. Bound. Elem. 88, 156–169 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  15. Cobos Sánchez, C., García-Pacheco, F.J., Moreno-Pulido, S., Sáez-Martínez, S.: Supporting vectors of continuous linear operators. Ann. Funct. Anal. 8(4), 520–530 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  16. Cobos-Sánchez, C., García-Pacheco, F.J., Guerrero-Rodríguez, J.M., García-Barrachina, L.: Solving an IBEM with supporting vector analysis to design quiet TMS coils. Eng. Anal. Bound. Elem. 117, 1–12 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  17. Diestel, J.: Sequences and Series in Banach Spaces. Graduate Texts in Mathematics. Springer, New York (1984)

    Book  Google Scholar 

  18. Drewnowski, L., Lipecki, Z.: On vector measures which have everywhere infinite variation or noncompact range. Dissertationes Math. (Rozprawy Mat.) 339, 39 (1995)

    MathSciNet  MATH  Google Scholar 

  19. García-Pacheco, F.J.: Banach spaces with an infinite number of smooth faces in their unit ball. J. Convex Anal. 15(2), 215–218 (2008)

    MathSciNet  MATH  Google Scholar 

  20. García-Pacheco, F.J.: Vector subspaces of the set of non-norm-attaining functionals. Bull. Aust. Math. Soc. 77(3), 425–432 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. García-Pacheco, F.J.: Convex components and multi-slices in real topological vector spaces. Ann. Funct. Anal. 6(3), 73–86 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. García-Pacheco, F.J.: Selfadjoint operators on real or complex Banach spaces. Nonlinear Anal. 192, 111696 (2020). 12 pp

    Article  MathSciNet  MATH  Google Scholar 

  23. García-Pacheco, F.J., Cobos-Sánchez, C., Moreno-Pulido, S., Sánchez-Alzola, A.: Exact solutions to \(\max _{\Vert x\Vert =1} \sum _{n=1}^\infty \Vert T_i(x)\Vert ^2\) with applications to physics, bioengineering and statistics. Commun. Nonlinear Sci. Numer. Simul. 82, 105054 (2020)

    MathSciNet  MATH  Google Scholar 

  24. García-Pacheco, F.J., Miralles, A.: Real renormings on complex Banach spaces. Chin. Ann. Math. Ser. B 29(3), 239–246 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. García-Pacheco, F.J., Naranjo-Guerra, E.: Supporting vectors of continuous linear projections. Int. J. Funct. Anal. Oper. Theory Appl. 9, 85–95 (2017)

    MATH  Google Scholar 

  26. García-Pacheco, F.J., Palmberg, N., Seoane-Sepúlveda, J.B.: Lineability and algebrability of pathological phenomena in analysis. J. Math. Anal. Appl. 326(2), 929–939 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  27. García-Pacheco, F.J., Pérez-Eslava, C., Seoane-Sepúlveda, J.B.: Moduleability, algebraic structures, and nonlinear properties. J. Math. Anal. Appl. 370(1), 159–167 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  28. García-Pacheco, F.J., Puglisi, D.: Lineability of functionals and operators. Studia Math. 201(1), 37–47 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  29. García-Pacheco, F.J., Puglisi, D.: A short note on the lineability of norm-attaining functionals in subspaces of \(\ell _\infty \). Arch. Math. (Basel) 105(5), 461–465 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  30. García-Pacheco, F.J., Puglisi, D.: Renormings concerning the lineability of the norm-attaining functionals. J. Math. Anal. Appl. 445(2), 1321–1327 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  31. García-Pacheco, F.J., Puglisi, D.: Lineability of functionals and renormings. Bull. Belg. Math. Soc. Simon Stevin. 25(1), 141–147 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  32. García-Pacheco, F.J., Rambla-Barreno, F., Seoane-Sepúlveda, J.B.: \(\mathbb{Q}\)-linear functions, functions with dense graph, and everywhere surjectivity. Math. Scand. 102(1), 156–160 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  33. García-Pacheco, F.J., Sáez-Martínez, S.: Normalizing rings. Banach J. Math. Anal. 14(3), 1143–1176 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  34. García-Pacheco, F.J., Seoane-Sepúlveda, J.B.: Vector spaces of non-measurable functions. Acta Math. Sin. (Engl. Ser.) 22(6), 1805–1808 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  35. Gurariy, V.I.: Linear spaces composed of everywhere nondifferentiable functions. C. R. Acad. Bulgare. Sci. 44, 13–16 (1991)

    MathSciNet  Google Scholar 

  36. Gurariy, V.I., Quarta, L.: On lineability of sets of continuous functions. J. Math. Anal. Appl. 294, 62–72 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  37. Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms. I. Fundamentals. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 305. Springer, Berlin (1993)

    MATH  Google Scholar 

  38. James, R.C.: Reflexivity and the supremum of linear functionals. Ann. Math. 2(66), 159–169 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  39. Kadets, V., López, G., Martín, M., Werner, D.: Equivalent norms with an extremely nonlineable set of norm attaining functionals. J. Inst. Math. Jussieu 19(1), 259–279 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  40. Levine, B., Milman, D.: On linear sets in space \(C\) consisting of functions of bounded variation. Commun. Inst. Sci. Math. Méc. Univ. Kharkoff [Zapiski Inst. Mat. Mech.] (4) 16, 102–105 (1940)

    MathSciNet  Google Scholar 

  41. Lindenstrauss, J.: On operators which attain their norm. Israel J. Math. 1, 139–148 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  42. Martín, M.: On proximinality of subspaces and the lineability of the set of norm-attaining functionals of Banach spaces. J. Funct. Anal. 278(4), 108353 (2020). 14 pp

    Article  MathSciNet  MATH  Google Scholar 

  43. Megginson, R.E.: An Introduction to Banach Space Theory, Graduate Texts in Mathematics. Springer, New York (1998)

    Book  MATH  Google Scholar 

  44. Moreno-Pulido, S., García-Pacheco, F.J., Cobos-Sánchez, C., Sánchez-Alzola, A.: Exact Solutions to the Maxmin Problem \( \max \Vert Ax\Vert \) Subject to \(\Vert Bx\Vert \le 1\). Mathematics 8(1), 85 (2020)

    Article  Google Scholar 

  45. Partington, J.R.: Norm attaining operators. Israel J. Math. 43(3), 273–276 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  46. Read, C.J.: Banach spaces with no proximinal subspaces of codimension \(2\). Israel J. Math. 223, 493–504 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  47. Rmoutil, M.: Norm-attaining functionals need not contain \(2\)-dimensional subspaces. J. Funct. Anal. 272, 918–928 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  48. Warner, S.: Topological fields. North-Holland Mathematics Studies, 157. In: Notas de Matemtica [Mathematical Notes], vol. 126. North-Holland Publishing Co., Amsterdam (1989)

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Acknowledgements

The author would like to express his deepest gratitude towards the referees for their valuable comments, suggestions and remarks that help improve the manuscript considerably.

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  1. Department of Mathematics, College of Engineering, University of Cádiz, Puerto Real, 11519, Spain

    Francisco Javier García-Pacheco

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  1. Francisco Javier García-Pacheco
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The author has been partially supported by Research Grant PGC-101514-B-I00 awarded by the Ministry of Science, Innovation and Universities of Spain. This work has been co-financed by the 2014-2020 ERDF Operational Programme and by the Department of Economy, Knowledge, Business and University of the Regional Government of Andalusia. Project reference: FEDER-UCA18-105867.

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García-Pacheco, F.J. Lineability of the set of supporting vectors. RACSAM 115, 41 (2021). https://doi.org/10.1007/s13398-020-00981-6

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