Topologies and bornologies determined by operator ideals, II

Wong, Ngai-Ching

Studia Mathematica, Tome 108 (1994), p. 153-162 / Harvested from The Polish Digital Mathematics Library

Résumé

Let be an operator ideal on LCS’s. A continuous seminorm p of a LCS X is said to be - continuous if $Q ̃_{p} \in^{i n j} (X, X ̃_{p})$ , where $X ̃_{p}$ is the completion of the normed space $X_{p} = X / p^{- 1} (0)$ and $Q ̃_{p}$ is the canonical map. p is said to be a Groth()- seminorm if there is a continuous seminorm q of X such that p ≤ q and the canonical map $Q ̃_{p q} : X ̃_{q} \to X ̃_{p}$ belongs to $(X ̃_{q}, X ̃_{p})$ . It is well known that when is the ideal of absolutely summing (resp. precompact, weakly compact) operators, a LCS X is a nuclear (resp. Schwartz, infra-Schwartz) space if and only if every continuous seminorm p of X is -continuous if and only if every continuous seminorm p of X is a Groth()-seminorm. In this paper, we extend this equivalence to arbitrary operator ideals and discuss several aspects of these constructions which were initiated by A. Grothendieck and D. Randtke, respectively. A bornological version of the theory is also obtained.

Publié le : 1994-01-01

Zbl 0805.46002

EUDML-ID : urn:eudml:doc:216125

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     title = {Topologies and bornologies determined by operator ideals, II},
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     year = {1994},
     pages = {153-162},
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Wong, Ngai-Ching. Topologies and bornologies determined by operator ideals, II. Studia Mathematica, Tome 108 (1994) pp. 153-162. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv111i2p153bwm/

Bibliographie

[00000] [1] L. Franco and C. Piñeiro, The injective hull of an operator ideal on locally convex spaces, Manuscripta Math. 38 (1982), 333-341. | Zbl 0531.47036

[00001] [2] H. Hogbe-Nlend, Bornologies and Functional Analysis, Math. Stud. 26, North-Holland, Amsterdam, 1977.

[00002] [3] H. Hogbe-Nlend, Nuclear and Co-Nuclear Spaces, Math. Stud. 52, North-Holland, Amsterdam, 1981.

[00003] [4] G. J. O. Jameson, Summing and Nuclear Norms in Banach Space Theory, London Math. Soc. Students Text 8, Cambridge University Press, Cambridge, 1987. | Zbl 0634.46007

[00004] [5] H. Jarchow, Locally Convex Spaces, Teubner, Stuttgart, 1981. | Zbl 0466.46001

[00005] [6] H. Jarchow, On certain locally convex topologies on Banach spaces, in: Functional Analysis: Surveys and Recent Results, III, K. D. Bierstedt and B. Fuchssteiner (eds.), North-Holland, Amsterdam, 1984, 79-93.

[00006] [7] H. Junek, Locally Convex Spaces and Operator Ideals, Teubner-Texte Math. 56, Teubner, Leipzig, 1983.

[00007] [8] M. Lindstorm, A characterization of Schwartz spaces, Math. Z. 198 (1988), 423-430. | Zbl 0653.46007

[00008] [9] A. Pietsch, Nuclear Locally Convex Spaces, Springer, Berlin, 1972.

[00009] [10] A. Pietsch, Operator Ideals, North-Holland, Amsterdam, 1980.

[00010] [11] D. Randtke, Characterizations of precompact maps, Schwartz spaces and nuclear spaces, Trans. Amer. Math. Soc. 165 (1972), 87-101. | Zbl 0209.14405

[00011] [12] H. H. Schaefer, Topological Vector Spaces, Springer, Berlin, 1971.

[00012] [13] I. Stephani, Injektive Operatorenideale über der Gesamtheit aller Banachräume und ihre topologische Erzeugung, Studia Math. 38 (1970), 105-124. | Zbl 0206.13104

[00013] [14] I. Stephani, Surjektive Operatorenideale über der Gesamtheit aller Banachräume, Wiss. Z. Friedrich-Schiller-Univ. Jena, Math.-Natur. Reihe 21 (1972), 187-216.

[00014] [15] I. Stephani, Surjektive Operatorenideale über der Gesamtheit aller Banachräume und ihre Erzeugung, Beiträge Anal. 5 (1973), 75-89.

[00015] [16] I. Stephani, Generating system of sets and quotients of surjective operator ideals, Math. Nachr. 99 (1980), 13-27. | Zbl 0474.47019

[00016] [17] I. Stephani, Generating topologies and quotients of injective operator ideals, in: Banach Space Theory and Its Applications (Proc., Bucharest 1981), Lecture Notes in Math. 991, Springer, Berlin, 1983, 239-255.

[00017] [18] N.-C. Wong and Y.-C. Wong, Bornologically surjective hull of an operator ideal on locally convex spaces, Math. Nachr. 160 (1993), 265-275. | Zbl 0810.47046

[00018] [19] Y.-C. Wong, The Topology of Uniform Convergence on Order-Bounded Sets, Lecture Notes in Math. 531, Springer, Berlin, 1976.

[00019] [20] Y.-C. Wong, Schwartz Spaces, Nuclear Spaces and Tensor Products, Lecture Notes in Math. 726, Springer, Berlin, 1979. | Zbl 0413.46001

[00020] [21] Y.-C. Wong and N.-C. Wong, Topologies and bornologies determined by operator ideals, Math. Ann. 282 (1988), 587-614. | Zbl 0633.46006