Annals of Functional Analysis

Some trace monotonicity properties and applications

Jean-Michel Combes and Peter D. Hislop

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Abstract

We present some results on the monotonicity of some traces involving functions of self-adjoint operators with respect to the natural ordering of their associated quadratic forms. The relation between these results and Löwner’s Theorem is discussed. We also apply these results to complete a proof of the Wegner estimate for continuum models of random Schrödinger operators as given in a 1994 paper by Combes and Hislop.

Article information

Source
Ann. Funct. Anal. Volume 7, Number 3 (2016), 394-401.

Dates
Received: 3 August 2015
Accepted: 1 December 2015
First available in Project Euclid: 19 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.afa/1463684085

Digital Object Identifier
doi:10.1215/20088752-3605258

Mathematical Reviews number (MathSciNet)
MR3506611

Zentralblatt MATH identifier
1351.47013

Subjects
Primary: 47B10: Operators belonging to operator ideals (nuclear, p-summing, in the Schatten-von Neumann classes, etc.) [See also 47L20]
Secondary: 47A60: Functional calculus 47B80: Random operators [See also 47H40, 60H25]

Keywords
operator trace inequalities operator monotone functions Loewner’s Theorem random Schrodinger operators Wegner estimate

Citation

Combes, Jean-Michel; Hislop, Peter D. Some trace monotonicity properties and applications. Ann. Funct. Anal. 7 (2016), no. 3, 394--401. doi:10.1215/20088752-3605258. https://projecteuclid.org/euclid.afa/1463684085


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References

  • [1] J. Bendat and S. Sherman, Monotone and convex operator functions, Trans. American Math. Soc. 79 (1955) 58–71.
  • [2] R. Bhatia and K. Sinha, Variation of real powers of positive operators, Indiana Univ. Math. J. 43 (1994), no. 3, 913–925.
  • [3] P. Chansangiam, Operator monotone functions: Characterizations and integral representations, preprint, arXiv:1305.2471v1.
  • [4] J.-M. Combes and P. D. Hislop, Localization for some continuous, random Hamiltonians in $d$-dimensions, J. Funct. Anal. 124 (1994), no. 1, 149–180.
  • [5] J.-M. Combes, P. D. Hislop, and F. Klopp, An optimal Wegner estimate and its application to the global continuity of the integrated density of states for random Schrödinger operators, Duke Math. J. 140 (2007), no. 3, 469–498.
  • [6] W. Donoghue, Monotone Matrix Functions and Analytic Continuation, Grundlehren Math. Wiss. 207, Springer, New York, 1974.
  • [7] F. Hansen, The fast track to Löwner’s theorem, Linear Algebra Appl. 438 (2013), no. 11, 4557–4571.
  • [8] T. Kato, Perturbation Theory for Linear Operators, Classics Math., Springer, Berlin, 1995.
  • [9] K. Löwner, Über monotone matrixfunktionen, Math. Z. 38 (1934), no. 1, 177–216.
  • [10] M. Reed and B. Simon, Methods of Modern Mathematical Physics IV: Analysis of Operators, Academic, New York, 1978.