Skip to main content
Log in

Abstract

The spacec p is the class of operators on a Hilbert space for which thec p norm |T| p =[trace(T*T)p/2]1/p is finite. We prove many of the known results concerningc p in an elementary fashion, together with the result (new for 1<p<2) thatc p is as uniformly convex a Banach space asl p. In spite of the remarkable parallel of norm inequalities in the spacesc p andl p, we show thatp ≠ 2, noc p built on an infinite dimensional Hilbert space is equivalent to any subspace of anyl p orL p space.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. J. A. Clarkson.Uniformly convex spaces, Trans. Amer. Math. Soc.40 (1936), 396–414.

    Article  MATH  MathSciNet  Google Scholar 

  2. J. Dixmier,Formes linéaires sur un anneau d’operateurs, Bull. Soc. Math. France81 (1953) 9–39.

    MATH  MathSciNet  Google Scholar 

  3. N. Dunford,Spectral Operators, Pacific J. Math.4 (1954), 321–354.

    MATH  MathSciNet  Google Scholar 

  4. N. Dunford and J. T. Schwartz.Linear Operators, Part II. Interscience, New York (1963).

    MATH  Google Scholar 

  5. I. C. Gohberg and M. G. Krein.Introduction to the Theory of Linear Non-Self-Adjoint Operators in Hilbert Space, Moscow (1965). (In Russian).

  6. A. Grothendieck,Produits tensoriels topologiques et espaces nucléaires, Memoirs Amer. Math. Soc.16 (1955).

  7. A. Horn,On the singular values of a product of completely continuous operators, Proc. Nat. Acad. Sci. U.S.A.36 (1950), 373–375.

    MathSciNet  Google Scholar 

  8. S. Kakutani,An example concerning uniform boundedness of spectral measures, Pacific J. Math.4 (1954), 363–372.

    MATH  MathSciNet  Google Scholar 

  9. W. Littman, C. McCarthy, and N. Rivière,L p Multiplier theorems of Marcinkiewicz type, To appear.

  10. C. McCarthy,Commuting Boolean algebras of projections II, Proc. Amer. Math. Soc.15 (1964), 781–787.

    Article  MATH  MathSciNet  Google Scholar 

  11. J. von Neumann.Some matrix-inequalities and metrization of matric-space, Tomsk Univ. Rev.1 (1937), 286–300.

    Google Scholar 

  12. B. J. Pettis.A proof that every uniformly convex space is reflexive, Duke Math. J.5 (1939), 249–253

    Article  MATH  MathSciNet  Google Scholar 

  13. R. Schatten,A theory of cross-spaces, Ann. of Math. Studies, No. 26, Princeton University Press, Princeton, 1950.

    MATH  Google Scholar 

  14. --,Norm ideals of completely continuous operators, Ergebnisse der Math, Neue Folge,27, Springer Verlag, 1960.

Download references

Author information

Authors and Affiliations

Authors

Additional information

The author was supported by National Science Foundation Grant GP-5707.

Rights and permissions

Reprints and permissions

About this article

Cite this article

McCarthy, C.A. Cp . Israel J. Math. 5, 249–271 (1967). https://doi.org/10.1007/BF02771613

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02771613

Keywords

Navigation