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Positivity

, Volume 12, Issue 2, pp 221–240 | Cite as

Delta-semidefinite and Delta-convex Quadratic Forms in Banach Spaces

  • Nigel Kalton
  • Sergei V. Konyagin
  • Libor Veselý
Article

Abstract

A continuous quadratic form (“quadratic form”, in short) on a Banach space X is: (a) delta-semidefinite (i.e., representable as a difference of two nonnegative quadratic forms) if and only if the corresponding symmetric linear operator \(T: X\rightarrow X^*\) factors through a Hilbert space; (b) delta-convex (i.e., representable as a difference of two continuous convex functions) if and only if T is a UMD-operator. It follows, for instance, that each quadratic form on an infinite-dimensional L p (μ) space (1 ≤ p ≤ ∞) is: (a) delta-semidefinite iff p ≥ 2; (b) delta-convex iff p > 1. Some other related results concerning delta-convexity are proved and some open probms are stated.

Mathematics Subject Classification (2000)

Primary 46B99 Secondary 52A41, 15A63 

Keywords

Banach space continuous quadratic form positively semidefinite quadratic form delta-semidefinite quadratic form delta-convex function Walsh-Paley martingale 

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Copyright information

© Springer Science + Business Media B.V. 2008

Authors and Affiliations

  • Nigel Kalton
    • 1
  • Sergei V. Konyagin
    • 2
  • Libor Veselý
    • 3
  1. 1.Department of MathematicsUniversity of Missouri-ColumbiaColumbiaU.S.A.
  2. 2.Department of Mechanics and MathematicsMoscow State UniversityMoscowRussia
  3. 3.Dipartimento di MatematicaUniversità degli Studi di MilanoMilanoItaly

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