Topologies and bornologies determined by operator ideals, II

Ngai-Ching Wong

ArticleOriginal scientific text

Title

Topologies and bornologies determined by operator ideals, II

Authors ¹

Affiliations

Department of Applied Mathematics, National Sun Yat-sen University, Kao-hsiung, 80424, Taiwan, R.O.C.

Abstract

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^{\in j} (X, X ̃_{p})

, where

X ̃_{p}

is the completion of the normed space

X_{p} = \frac{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.

Keywords

ENG

operator ideals, locally convex spaces, topologies, bornologies, Grothendieck spaces

L. Franco and C. Piñeiro, The injective hull of an operator ideal on locally convex spaces, Manuscripta Math. 38 (1982), 333-341.
H. Hogbe-Nlend, Bornologies and Functional Analysis, Math. Stud. 26, North-Holland, Amsterdam, 1977.
H. Hogbe-Nlend, Nuclear and Co-Nuclear Spaces, Math. Stud. 52, North-Holland, Amsterdam, 1981.
G. J. O. Jameson, Summing and Nuclear Norms in Banach Space Theory, London Math. Soc. Students Text 8, Cambridge University Press, Cambridge, 1987.
H. Jarchow, Locally Convex Spaces, Teubner, Stuttgart, 1981.
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.
H. Junek, Locally Convex Spaces and Operator Ideals, Teubner-Texte Math. 56, Teubner, Leipzig, 1983.
M. Lindstorm, A characterization of Schwartz spaces, Math. Z. 198 (1988), 423-430.
A. Pietsch, Nuclear Locally Convex Spaces, Springer, Berlin, 1972.
A. Pietsch, Operator Ideals, North-Holland, Amsterdam, 1980.
D. Randtke, Characterizations of precompact maps, Schwartz spaces and nuclear spaces, Trans. Amer. Math. Soc. 165 (1972), 87-101.
H. H. Schaefer, Topological Vector Spaces, Springer, Berlin, 1971.
I. Stephani, Injektive Operatorenideale über der Gesamtheit aller Banachräume und ihre topologische Erzeugung, Studia Math. 38 (1970), 105-124.
I. Stephani, Surjektive Operatorenideale über der Gesamtheit aller Banachräume, Wiss. Z. Friedrich-Schiller-Univ. Jena, Math.-Natur. Reihe 21 (1972), 187-216.
I. Stephani, Surjektive Operatorenideale über der Gesamtheit aller Banachräume und ihre Erzeugung, Beiträge Anal. 5 (1973), 75-89.
I. Stephani, Generating system of sets and quotients of surjective operator ideals, Math. Nachr. 99 (1980), 13-27.
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.
N.-C. Wong and Y.-C. Wong, Bornologically surjective hull of an operator ideal on locally convex spaces, Math. Nachr. 160 (1993), 265-275.
Y.-C. Wong, The Topology of Uniform Convergence on Order-Bounded Sets, Lecture Notes in Math. 531, Springer, Berlin, 1976.
Y.-C. Wong, Schwartz Spaces, Nuclear Spaces and Tensor Products, Lecture Notes in Math. 726, Springer, Berlin, 1979.
Y.-C. Wong and N.-C. Wong, Topologies and bornologies determined by operator ideals, Math. Ann. 282 (1988), 587-614.

Title

Topologies and bornologies determined by operator ideals, II

Affiliations

Abstract

Keywords

Bibliography