Topologies and bornologies determined by operator ideals, II

Ngai-Ching Wong

Studia Mathematica (1994)

  • Volume: 111, Issue: 2, page 153-162
  • ISSN: 0039-3223

Abstract

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Let be an operator ideal on LCS’s. A continuous seminorm p of a LCS X is said to be - continuous if Q ̃ p ∈ 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 → 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.

How to cite

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Wong, Ngai-Ching. "Topologies and bornologies determined by operator ideals, II." Studia Mathematica 111.2 (1994): 153-162. <http://eudml.org/doc/216125>.

@article{Wong1994,
abstract = {Let be an operator ideal on LCS’s. A continuous seminorm p of a LCS X is said to be - continuous if $Q̃_p ∈ ^\{inj\}(X,X̃_p)$, where $X̃_p$ is the completion of the normed space $X_p = X/p^\{-1\}(0)$ and $Q̃_p$ 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 $Q̃_\{pq\} : X̃_q → X̃_p$ belongs to $(X̃_q,X̃_p)$. 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.},
author = {Wong, Ngai-Ching},
journal = {Studia Mathematica},
keywords = {operator ideals; locally convex spaces; topologies; bornologies; Grothendieck spaces; ideal of absolutely summing operators; nuclear space; operator ideal},
language = {eng},
number = {2},
pages = {153-162},
title = {Topologies and bornologies determined by operator ideals, II},
url = {http://eudml.org/doc/216125},
volume = {111},
year = {1994},
}

TY - JOUR
AU - Wong, Ngai-Ching
TI - Topologies and bornologies determined by operator ideals, II
JO - Studia Mathematica
PY - 1994
VL - 111
IS - 2
SP - 153
EP - 162
AB - Let be an operator ideal on LCS’s. A continuous seminorm p of a LCS X is said to be - continuous if $Q̃_p ∈ ^{inj}(X,X̃_p)$, where $X̃_p$ is the completion of the normed space $X_p = X/p^{-1}(0)$ and $Q̃_p$ 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 $Q̃_{pq} : X̃_q → X̃_p$ belongs to $(X̃_q,X̃_p)$. 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.
LA - eng
KW - operator ideals; locally convex spaces; topologies; bornologies; Grothendieck spaces; ideal of absolutely summing operators; nuclear space; operator ideal
UR - http://eudml.org/doc/216125
ER -

References

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  12. [12] H. H. Schaefer, Topological Vector Spaces, Springer, Berlin, 1971. 
  13. [13] I. Stephani, Injektive Operatorenideale über der Gesamtheit aller Banachräume und ihre topologische Erzeugung, Studia Math. 38 (1970), 105-124. Zbl0206.13104
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  16. [16] I. Stephani, Generating system of sets and quotients of surjective operator ideals, Math. Nachr. 99 (1980), 13-27. Zbl0474.47019
  17. [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. 
  18. [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. Zbl0810.47046
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  21. [21] Y.-C. Wong and N.-C. Wong, Topologies and bornologies determined by operator ideals, Math. Ann. 282 (1988), 587-614. Zbl0633.46006

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