Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-26T09:47:05.404Z Has data issue: false hasContentIssue false
  • English
  • Français

Some criteria for nuclearity

Published online by Cambridge University Press:  24 October 2008

M. A. Sofi
M. A. Sofi
Affiliation:
Department of Mathematics, Aligarh Muslim University, Aligarh 202001, India
Get access Rights & Permissions [Opens in a new window]

Extract

For a given locally convex space, it is always of interest to find conditions for its nuclearity. Well known results of this kind – by now already familiar – involve the use of tensor products, diametral dimension, bilinear forms, generalized sequence spaces and a host of other devices for the characterization of nuclear spaces (see [9]). However, it turns out, these nuclearity criteria are amenable to a particularly simple formulation in the setting of certain sequence spaces; an elegant example is provided by the so-called Grothendieck–Pietsch (GP, for short) criterion for nuclearity of a sequence space (in its normal topology) in terms of the summability of certain numerical sequences.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 100 , Issue 1 , July 1986 , pp. 151 - 159
Copyright
Copyright © Cambridge Philosophical Society 1986

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Crone, L., Fleming, D. J. and Jessup, P.. Diagonal nuclear operators. Studia Math. 43 (1972), 52–56.CrossRefGoogle Scholar
[2]Dubinsky, E. and Ramanujan, M. S.. On λ-nuclearity. Mem. Amer. Math. Soc. 128 (1972).Google Scholar
[3]Dubinsky, E. and Ramanujan, M. S.. Λ(α)-bases and nuclear spaces. J. Indian Math. Soc. 36 (1972), 333–345.Google Scholar
[4]Dubinsky, E.. The Structure of Nuclear Fréchet Spaces. Lecture Notes in Math. vol. 720 (Springer-Verlag, 1979).CrossRefGoogle Scholar
[5]Kamthan, P. and Gupta, M.. Sequence Spaces and Series. Lecture Notes, 65 (Marcel Dekker Inc., 1981).Google Scholar
[6]Kamthan, P. K., Gupta, M. and Sofi, M. A.. λ-bases and their applications. J. Math. Anal. Appl. 88 (1981), 76–99.CrossRefGoogle Scholar
[7]Kamthan, P. K. and Sofi, M. A.. λ-bases and λ-nuclearity. J. Math. Anal. Appl. 99 (1984), 164–188.CrossRefGoogle Scholar
[8]Nelimarkka, E.. On operator ideals and locally convex -spaces with applications to λ-nuclearity. Ann. Acad. Sci. Fenn. A.I. Math. Dissertationes 13 (1977).Google Scholar
[9]Pietsch, A.. Nuclear Locally Convex Spaces (Springer-Verlag, 1972).Google Scholar
[10]Pietsch, A.. Operator Ideals. (North-Holland, 1980).Google Scholar
[11]Ramanujan, M. S. and Rosenberger, B.. On λ(ø P)-nuclearity. Comp. Math. 34(2), (1977), 113–125.Google Scholar
[12]Ramanujan, M. S. and Schock, E.. Nuclearity, bases and finite dimensional decompositions. (Preprint.)Google Scholar
[13]Ramanujan, M. S. and Terzioglu, T.. Diametral dimensions of cartesian products, stability of smooth sequence spaces and applications. J. Reine Angew. Math. 280 (1976), 163–171.Google Scholar
[14]Ruckle, W. H.. Topologies on sequence spaces. Pacific J. Math. 42 (1972), 235–249.CrossRefGoogle Scholar
[15]Ruckle, W. H.Sequence Spaces. Research Notes in Mathematics 49 (Pitman, 1981).Google Scholar
[16]Wojtyński, W.. On conditional bases in non-nuclear Fréchet-spaces. Studia Math. 35 (1970), 77–96.CrossRefGoogle Scholar
[17]Garcia, J. M. L.. ΛN(P)-nuclearity and basis. Math. Nachr. 121 (1985), 7–10.Google Scholar