Nuclear tight operators on spaces of continuous functions

Nowak, Marian

doi:10.1090/proc/15911

Nuclear tight operators on spaces of continuous functions
HTML articles powered by AMS MathViewer

by Marian Nowak PDF

Proc. Amer. Math. Soc. 150 (2022), 3487-3500 Request permission

Abstract:

Let $C_b(X)$ denote the Banach space of all bounded continuous real-valued functions on a completely regular Hausdorff space $X$. We characterize nuclear tight operators $T:C_b(X)\rightarrow E$ in terms of their representing Radon vector measures. As an application, it is shown that some natural kernel operators on $C_b(X)$ are tight and nuclear.

References

José Aguayo and José Sánchez, Weakly compact operators and the strict topologies, Bull. Austral. Math. Soc. 39 (1989), no. 3, 353–359. MR 995133, DOI 10.1017/S0004972700003270
Charalambos D. Aliprantis and Owen Burkinshaw, Positive operators, Pure and Applied Mathematics, vol. 119, Academic Press, Inc., Orlando, FL, 1985. MR 809372
Charalambos D. Aliprantis and Owen Burkinshaw, Locally solid Riesz spaces with applications to economics, 2nd ed., Mathematical Surveys and Monographs, vol. 105, American Mathematical Society, Providence, RI, 2003. MR 2011364, DOI 10.1090/surv/105
K. P. S. Bhaskara Rao and M. Bhaskara Rao, Theory of charges, Pure and Applied Mathematics, vol. 109, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. A study of finitely additive measures; With a foreword by D. M. Stone. MR 751777
John B. Conway, A course in functional analysis, 2nd ed., Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, 1990. MR 1070713
Romulus Cristescu, Topological vector spaces, Editura Academiei, Bucharest; Noordhoff International Publishing, Leyden, 1977. Translated from the Romanian by Mihaela Suliciu. MR 0454552
J. Diestel and J. J. Uhl Jr., Vector measures, Mathematical Surveys, No. 15, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis. MR 0453964, DOI 10.1090/surv/015
N. Dinculeanu, Vector measures, International Series of Monographs in Pure and Applied Mathematics, Vol. 95, Pergamon Press, Oxford-New York-Toronto, Ont.; VEB Deutscher Verlag der Wissenschaften, Berlin, 1967. MR 0206190
Nicolae Dinculeanu, Vector integration and stochastic integration in Banach spaces, Pure and Applied Mathematics (New York), Wiley-Interscience, New York, 2000. MR 1782432, DOI 10.1002/9781118033012
Israel Gohberg, Seymour Goldberg, and Nahum Krupnik, Traces and determinants of linear operators, Operator Theory: Advances and Applications, vol. 116, Birkhäuser Verlag, Basel, 2000. MR 1744872, DOI 10.1007/978-3-0348-8401-3
A. Grothendieck, Résumé de la théorie métrique des produits tensoriels topologiques, Bol. Soc. Mat. São Paulo 8 (1953), 1–79 (French). MR 94682
A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Séminaire Bourbaki, Vol. 2, Soc. Math. France, Paris, 1995, pp. Exp. No. 69, 193–200 (French). MR 1609222
Hans Jarchow, Locally convex spaces, Mathematische Leitfäden. [Mathematical Textbooks], B. G. Teubner, Stuttgart, 1981. MR 632257, DOI 10.1007/978-3-322-90559-8
Marian Nowak, Weakly compact operators and strict topologies, Topology Appl. 160 (2013), no. 16, 2053–2060. MR 3106460, DOI 10.1016/j.topol.2013.08.011
Marian Nowak, Compactness and sequential completeness in some spaces of operators, Positivity 18 (2014), no. 2, 359–373. MR 3215183, DOI 10.1007/s11117-013-0248-2
Marian Nowak, A Riesz representation theory for completely regular Hausdorff spaces and its applications, Open Math. 14 (2016), no. 1, 474–496. MR 3530565, DOI 10.1515/math-2016-0043
Marian Nowak, Strongly Mackey topologies and Radon vector measures, Port. Math. 77 (2020), no. 3-4, 283–297. MR 4191643, DOI 10.4171/pm/2052
Albrecht Pietsch, Nuclear locally convex spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 66, Springer-Verlag, New York-Heidelberg, 1972. Translated from the second German edition by William H. Ruckle. MR 0350360
A. Pietsch, Eigenvalues and $s$-numbers, Mathematik und ihre Anwendungen in Physik und Technik [Mathematics and its Applications in Physics and Technology], vol. 43, Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1987. MR 917067
Raymond A. Ryan, Introduction to tensor products of Banach spaces, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 2002. MR 1888309, DOI 10.1007/978-1-4471-3903-4
Helmut H. Schaefer, Topological vector spaces, Graduate Texts in Mathematics, Vol. 3, Springer-Verlag, New York-Berlin, 1971. Third printing corrected. MR 0342978, DOI 10.1007/978-1-4684-9928-5
F. Dennis Sentilles, Compactness and convergence in the space of measures, Illinois J. Math. 13 (1969), 761–768. MR 247447
F. Dennis Sentilles, Compact and weakly compact operators on $C(S)_{\beta }$, Illinois J. Math. 13 (1969), 769–776. MR 247448
M. A. Sofi, Vector measures and nuclear operators, Illinois J. Math. 49 (2005), no. 2, 369–383. MR 2163940
Alfred E. Tong, Nuclear mappings on $C(X)$, Math. Ann. 194 (1971), 213–224. MR 295136, DOI 10.1007/BF01350051
V. S. Varadarajan, Measures on topological spaces, Mat. Sb. (N.S.) 55 (97) (1961), 35–100 (Russian). MR 0148838
Robert F. Wheeler, A survey of Baire measures and strict topologies, Exposition. Math. 1 (1983), no. 2, 97–190. MR 710569
Kôsaku Yosida, Functional analysis, 4th ed., Die Grundlehren der mathematischen Wissenschaften, Band 123, Springer-Verlag, New York-Heidelberg, 1974. MR 0350358, DOI 10.1007/978-3-642-96208-0