Mathematica Bohemica

Mathematica Bohemica, first online, pp. 1-8

Absolutely summing operators on the Banach space of totally measurable functions

Marian Nowak

Received November 29, 2024. Published online June 5, 2025.

Abstract: Let $\Sigma$ be a $\sigma$-algebra of subsets of a set $\Omega$, and $X$ and $Y$ be Banach spaces. Let $B(\Sigma,X)$ stand for the Banach space of all $X$-valued totally measurable functions on $\Omega$, equipped with the supremum norm. We study absolutely summing operators ${T\colon B(\Sigma,X)\rightarrow Y}$. We characterize absolutely summing operators $T\colon B(\Sigma,X)\rightarrow Y$ in terms of their representing operator-valued measures. It is shown that the classes of dominated operators and absolutely summing operators $T\colon B(\Sigma,X)\rightarrow Y$ coincide if and only if every bounded linear operator $U\colon X\rightarrow Y$ is absolutely summing.

Keywords: space of totally measurable functions; dominated operator; absolutely summing operator; operator-valued measure

Classification MSC: 47B10, 46G10

DOI: 10.21136/MB.2025.0153-24

PDF available at: Institute of Mathematics CAS

References:
[1] F. Albiac, N. J. Kalton: Topics in Banach Space Theory. Graduate Texts in Mathematics 233. Springer, Cham (2016). DOI 10.1007/978-3-319-31557-7 | MR 3526021 | Zbl 1352.46002
[2] J. K. Brooks, P. W. Lewis: Operators on function spaces. Bull. Am. Math. Soc. 78 (1972), 697-701. DOI 10.1090/S0002-9904-1972-12988-4 | MR 0298442 | Zbl 0265.47021
[3] J. K. Brooks, P. W. Lewis: Linear operators and vector measures. Trans. Am. Math. Soc. 192 (1974), 139-162. DOI 10.1090/S0002-9947-1974-0338821-5 | MR 0338821 | Zbl 0331.46035
[4] A. Defant, K. Floret: Tensor Norms and Operator Ideals. North-Holland Mathematics Studies 176. North-Holland, Amsterdam (1993). DOI 10.1016/s0304-0208(08)x7019-7 | MR 1209438 | Zbl 0774.46018
[5] J. Diestel: An elementary characterization of absolutely summing operators. Math. Ann. 196 (1972), 101-105. DOI 10.1007/BF01419607 | MR 0306956 | Zbl 0221.46040
[6] J. Diestel: The Radon-Nikodym property and the coincidence of integral and nuclear operators. Rev. Roum. Math. Pures Appl. 17 (1972), 1611-1620. MR 0333728 | Zbl 0255.28010
[7] J. Diestel, H. Jarchow, A. Tonge: Absolutely Summing Operators. Cambridge Studies in Advanced Mathematics 43. Cambridge University Press, Cambridge (1995). DOI 10.1017/CBO9780511526138 | MR 1342297 | Zbl 0855.47016
[8] J. Diestel, J. J. Uhl, Jr.: Vector Measures. Mathematical Surveys 15. AMS, Providence (1977). DOI 10.1090/surv/015 | MR 0453964 | Zbl 0369.46039
[9] N. Dinculeanu: Vector Measures. Hochschulbücher für Mathematik 64. VEB Deutscher Verlag der Wissenschaften, Berlin (1966). DOI 10.1016/C2013-0-07847-4 | MR 0206189 | Zbl 0142.10502
[10] N. Dinculeanu: Vector Integration and Stochastic Integration in Banach Spaces. Pure and Applied Mathematics. A Wiley-Interscience Series of Texts, Monographs and Tracts. John Wiley and Sons, New York (2000). DOI 10.1002/9781118033012 | MR 1782432 | Zbl 0974.28006
[11] K. Floret, J. Wloka: Einführung in die Theorie der lokalconvexen Räume. Lectuer Notes in Mathematics 56. Springer, Berlin (1968). (In German.) DOI 10.1007/BFb0098549 | MR 0226355 | Zbl 0155.45101
[12] A. Grothendieck: Résumé de la théorie métrique des produits tensoriels topologiques. Bol. Soc. Mat. São Paulo 8 (1953), 1-79. (In French.) MR 0094682 | Zbl 0074.32303
[13] H. Jarchow: Locally Convex Spaces. Mathematische Leitfäden. B. G. Teubner, Stuttgart (1981). DOI 10.1007/978-3-322-90559-8 | MR 0632257 | Zbl 0466.46001
[14] J. Lindenstrauss, A. Pełczyński: Absolutely summing operators in $L_p$-spaces and their applications. Stud. Math. 29 (1968), 275-326. DOI 10.4064/sm-29-3-275-326 | MR 0231188 | Zbl 0183.40501
[15] M. Nowak: Decompositions of weakly compact operators on the space of totally measurable functions. Indag. Math., New Ser. 23 (2012), 381-387. DOI 10.1016/j.indag.2012.02.004 | MR 2948634 | Zbl 1259.47020
[16] A. Pietsch: Quasinukleare Abbildungen in normierten Räumen. Math. Ann. 165 (1966), 76-90. (In German.) DOI 10.1007/BF01351669 | MR 0198253 | Zbl 0171.12101
[17] A. Pietsch: Nuclear Locally Convex Spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete 66. Springer, Berlin (1972). DOI 10.1007/978-3-642-87665-3 | MR 0350360 | Zbl 0236.46001
[18] A. Pietsch: Eigenvalues and $s$-Numbers. Cambridge Studies in Advanced Mathematics 13. Cambridge University Press, Cambridge (1987). MR 0890520 | Zbl 0615.47019
[19] J. Rodríguez: Absolutely summing operators and integration of vector-valued functions. J. Math. Anal. Appl. 316 (2006), 579-600. DOI 10.1016/j.jmaa.2005.05.001 | MR 2207332 | Zbl 1097.46028
[20] A. H. Shuchat: Integral representation theorems in topological vector spaces. Trans. Am. Math. Soc. 172 (1972), 373-397. DOI 10.1090/S0002-9947-1972-0312264-0 | MR 0312264 | Zbl 0231.46079
[21] C. Swartz: Absolutely summing and dominated operators on spaces of vector-valued continuous functions. Trans. Am. Math. Soc. 179 (1973), 123-131. DOI 10.1090/S0002-9947-1973-0320796-5 | MR 0320796 | Zbl 0226.46038
[22] K. Swong: A representation theory of continuous linear maps. Math. Ann. 155 (1964), 270-291. DOI 10.1007/BF01354862 | MR 0165358 | Zbl 0197.10503

Affiliations: Marian Nowak, Institute of Mathematics, University of Zielona Góra, ul. Szafrana 4A, 65-516 Zielona Góra, Poland, e-mail: M.Nowak@wmie.uz.zgora.pl

[List of online first articles] [Contents of Mathematica Bohemica] [Full text of the older issues of Mathematica Bohemica at DML-CZ]

PDF available at:

Institute of Mathematics CAS
free access