The b-Gelfand–Phillips property for locally convex spaces

• Banakh, T. ^[2] ; Gabriyelyan, Saak ^[1]
  1. [1] Ben-Gurion University of the Negev

     Ben-Gurion University of the Negev

     Israel

     
  2. [2] Ivan Franko National University, Lviv, Ukraine
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 75, Fasc. 3, 2024, págs. 715-734
• Idioma: inglés
• DOI: 10.1007/s13348-023-00409-5
• Texto completo no disponible (Saber más ...)
• Resumen

  • We extend the well-known Gelfand–Phillips property for Banach spaces to locally convex spaces, defining a locally convex space E to be b-Gelfand–Phillips if every limited set in E, which is bounded in the strong topology \beta (E,E') on E, is precompact in \beta (E,E'). Several characterizations of b-Gelfand–Phillips spaces are given. The problem of preservation of the b-Gelfand–Phillips property by standard operations over locally convex spaces is considered. Also we explore the b-Gelfand–Phillips property in spaces C(X) of continuous functions on a Tychonoff space X. If \tau and {\mathcal{T}} are two locally convex topologies on C(X) such that {\mathcal{T}}_p\subseteq \tau \subseteq {\mathcal{T}}\subseteq {\mathcal{T}}_k, where {\mathcal{T}}_p is the topology of pointwise convergence and {\mathcal{T}}_k is the compact-open topology on C(X), then the b-Gelfand–Phillips property of the function space (C(X),\tau ) implies the b-Gelfand–Phillips property of (C(X),{\mathcal{T}}). If additionally X is metrizable, then the function space \big (C(X),{\mathcal{T}}\big ) is b-Gelfand–Phillips.

• Referencias bibliográficas
  • Alonso, C.: Limited sets in Fréchet spaces. Rev. Real Acad. Cienc. Exactas Fis. Nat. Madrid 89(1–2), 53–60 (1995)
  • Banakh, I., Banakh, T.: Constructing non-compact operators into c_0. Stud. Math. 201, 65–67 (2010)
  • Banakh, T., Gabriyelyan, S.: Banach spaces with the (strong) Gelfand–Phillips property. Banach J. Math. Anal. 16, art. 24 (2022)
  • Banakh, T., Gabriyelyan, S.: A simple Efimov space with sequentially-nice space of probability measures. Submitted. arXiv:2110.09062
  • Banakh, T., Gabriyelyan, S.: The Josefson–Nissenzweig property for locally convex spaces. Filomat 37(8), 2517–2529 (2023)
  • Bonet, J., Lindström, M., Valdivia, M.: Two theorems of Josefson–Nissenzweig type for Fréchet spaces. Proc. Am. Math. Soc. 117, 363–364 (1993)
  • Bourgain, J., Diestel, J.: Limited operators and strict cosingularity. Math. Nachr. 119, 55–58 (1984)
  • Castillo, J.M.F., González, M.: Three-Space Problems in Banach Space Theory. Springer Lecture Notes in Mathematics, vol. 1667, Springer-Verlag,...
  • Castillo, J.M.F., González, M., Papini, P.L.: On weak^\ast-extensible Banach spaces. Nonlinear Anal. 75, 4936–4941 (2012)
  • Correa, C., Tausk, D.: On extensions of c_0-valued operators. J. Math. Anal. Appl. 405(2), 400–408 (2013)
  • Correa, C., Tausk, D.: Compact lines and the Sobczyk property. J. Funct. Anal. 266(9), 5765–5778 (2014)
  • Diestel, J.: Sequences and Series in Banach Spaces. GTM, vol. 92. Springer, Berlin (1984)
  • Dorantes-Aldama, A., Shakhmatov, D.: Selective sequential pseudocompactness. Topol. Appl. 222, 53–69 (2017)
  • Drewnowski, L.: On Banach spaces with the Gelfand–Phillips property. Math. Z. 193, 405–411 (1986)
  • Drewnowski, L., Emmanuele, G.: On Banach spaces with the Gelfand–Phillips property, II. Rend. Circ. Mat. Palermo 38, 377–391 (1989)
  • Engelking, R.: General Topology. Heldermann Verlag, Berlin (1989)
  • Gabriyelayn, S.: Locally convex spaces and Schur type properties. Ann. Acad. Sci. Fenn. Math. 44, 363–378 (2019)
  • Gabriyelayn, S.: Locally convex spaces with Gelfand–Phillips type properties. Preprint
  • Gelfand, I.: Abstrakte Funktionen und lineare Operatoren. Mat. Sb. 4, 235–284 (1938)
  • Gillman, L., Jerison, M.: Rings of Continuous Functions. Van Nostrand, New York (1960)
  • Jarchow, H.: Locally Convex Spaces. B.G. Teubner, Stuttgart (1981)
  • Lindström, M., Schlumprecht, Th.: On limitedness in locally convex spaces. Arch. Math. 53, 65–74 (1989)
  • Phillips, R.S.: On linear transformations. Trans. Am. Math. Soc. 48, 516–541 (1940)
  • Ruess, W.: An operator space characterization of Fréchet spaces not containing \ell _1. Funct. Approx. Comment. Mat. 44(2), 259–264 (2011)
  • Schlumprecht, T.: Limited sets in Banach spaces. Dissertation, Univ. Munich (1987)
  • Schlumprecht, T.: Limited sets in C(K)-spaces and examples concerning the Gelfand–Phillips property. Math. Nachr. 157, 51–64 (1992)
  • Sinha, D.P., Arora, K.K.: On the Gelfand–Phillips property in Banach spaces with PRI. Collect. Math. 48(3), 347–354 (1997)