Ando’s theorem for nonnegative forms

Ando’s theorem for nonnegative forms In this paper, we present a generalization of Ando’s theorem for nonnegative forms. He proved that the infimum of two positive operators A and B exists in the positive cone if and only if the generalized shorts [B]A and [A]B are comparable (see Ando et al. in Problem of infimum in the positive cone, analytic and geometric inequalities and applications, Math. Appl. 478, pp 1–12, 1999). That is, [A]B ≤ [B]A or [B]A ≤ [A]B. Using the concept of the parallel sum of nonnegative forms, Hassi, Sebestyén and de Snoo investigated the decomposability of a nonnegative form $${\mathfrak{t}}$$ into an almost dominated and a singular part with respect to a nonnegative form $${\mathfrak{w}}$$ (see Hassi et al. in J. Funct. Anal. 257(12), 3858–3894, 2009). Applying their results, we formulate a necessary and sufficient condition for the existence of the infimum of two nonnegative forms. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Ando’s theorem for nonnegative forms

Positivity , Volume 16 (4) – Jun 21, 2011

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Publisher
Springer Journals
Copyright
Copyright © 2011 by Springer Basel AG
Subject
Mathematics; Potential Theory; Operator Theory; Fourier Analysis; Econometrics; Calculus of Variations and Optimal Control; Optimization
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-011-0133-9
Publisher site
See Article on Publisher Site

Abstract

In this paper, we present a generalization of Ando’s theorem for nonnegative forms. He proved that the infimum of two positive operators A and B exists in the positive cone if and only if the generalized shorts [B]A and [A]B are comparable (see Ando et al. in Problem of infimum in the positive cone, analytic and geometric inequalities and applications, Math. Appl. 478, pp 1–12, 1999). That is, [A]B ≤ [B]A or [B]A ≤ [A]B. Using the concept of the parallel sum of nonnegative forms, Hassi, Sebestyén and de Snoo investigated the decomposability of a nonnegative form $${\mathfrak{t}}$$ into an almost dominated and a singular part with respect to a nonnegative form $${\mathfrak{w}}$$ (see Hassi et al. in J. Funct. Anal. 257(12), 3858–3894, 2009). Applying their results, we formulate a necessary and sufficient condition for the existence of the infimum of two nonnegative forms.

Journal

PositivitySpringer Journals

Published: Jun 21, 2011

References

  • Shorted operators
    Anderson, W.N.
  • Infima of Hilbert space effects
    Moreland, T.; Gudder, S.

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