Hostname: page-component-848d4c4894-pjpqr Total loading time: 0 Render date: 2024-07-06T23:44:13.857Z Has data issue: false hasContentIssue false
  • English
  • Français

Biconcave-function characterisations of UMD and Hilbert spaces

Published online by Cambridge University Press:  17 April 2009

Jinsik Mok Lee
Jinsik Mok Lee
Affiliation:
Department of Mathematics, Sungwha University, 381-7 Samyong-Dong Chonahn 330-150, Republic of Korea
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Suppose that X is a real or complex Banach space with norm |·|. Then X is a Hilbert space if and only if

for all x in X and all X-valued Bochner integrable functions Y on the Lebesgue unit interval satisfying EY = 0 and |x − Y| ≤ 2 almost everywhere. This leads to the following biconcave-function characterisation: A Banach space X is a Hilbert space if and only if there is a biconcave function η: {(x, y) ∈ X × X: |x − y| ≤ 2} → R such that η(0, 0) = 2 and

If the condition η(0, 0) = 2 is eliminated, then the existence of such a function η characterises the class UMD (Banach spaces with the unconditionally property for martingale differences).

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 47 , Issue 2 , April 1993 , pp. 297 - 306
Copyright
Copyright © Australian Mathematical Society 1993

References

[1]Amir, D., Characterizations of inner product spaces, Operator theory: Advances and applications 20 (Birkhäuser Verlag, Boston, 1986).Google Scholar
[2]Burkholder, D.L., ‘A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional’, Ann. Probab. 9 (1981), 997–1011.CrossRefGoogle Scholar
[3]Burkholder, D.L., ‘Martingale transforms and the geometry of Banach spaces’, in Proceedings of the Third International Conference on Probability in Banach Spaces, Tufts University, 1980, Lecture Notes in Mathematics 860 (Springer-Verlag, Berlin, Heidelberg, New York, 1981), pp. 35–50.Google Scholar
[4]Burkholder, D.L., ‘Boundary value problems and sharp inequalities for martingale transforms’, Ann. Probab. 12 (1984), 647–702.Google Scholar
[5]Burkholder, D. L., ‘An elementary proof of an inequality of R.E.A.C. Paley’, Bull. London Math. Soc. 17 (1985), 474–478.CrossRefGoogle Scholar
[6]Burkholder, D.L., ‘An extension of a classical martingale inequality’, in Probability Theory and Harmonic Analysis, (Chao, J.A. and Woyczynski, W.A., Editors) (Marcel Dekker, New York (1986)), pp. 21–30.Google Scholar
[7]Burkholder, D.L., ‘Martingales and Fourier analysis in Banach spaces’, in C.I.M.E. Lectures, Varenna (Como), Italy, 1985, Lecture Notes in Math. 1206 (Springer-Verlag, Berlin, Heidelberg, New York, 1986), pp. 61–108.Google Scholar
[8]Burkholder, D.L., ‘Explorations in Martingale theory and its applications’, in Saint-Flour Lectures, 1989, Lecture Notes in Math. 1464 (Springer-Verlag, Berlin, Heidelberg, New York, 1991), pp. 1–66.Google Scholar
[9]Doob, J.L., Stochastic processes (Wiley, New York, 1953).Google Scholar
[10]Diestel, J. and Uhl, J.J., Vector measures, Math. Surveys 15 (Amer. Math. Soc., Providence, Rhode Island, 1977).Google Scholar
[11]Istraˇţescu, V.I., Inner product structures (D. Reidel Publishing Company, Boston, 1987).CrossRefGoogle Scholar