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Spectral dominance and commuting chains

Author(s): Bich T. Hoai; Charles R. Johnson; Ilya M. Spitkovsky
Journal: Proc. Amer. Math. Soc. 136 (2008), 2019-2029.
MSC (2000): Primary 47A63, 15A57, 15A27
Posted: February 14, 2008
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Abstract: A positive semidefinite (PSD) operator $ A$ ``spectrally dominates'' a PSD operator $ B$ if $ A^t-B^t$ is PSD for all $ t>0$. We (i) give a new characterization of spectral dominance in finite dimensions in terms of a monotonic chain of intermediate, pairwise commuting operators and (ii) determine for which pairs $ A,B$ spectral dominance persists under the taking of arbitrary compressions. Earlier results about spectral dominance are proven (in finite dimensions) in new ways, and several corollary observations are made.

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Additional Information:

Bich T. Hoai
Affiliation: Department of Mathematics, College of William & Mary, Williamsburg, Virginia 23185
Address at time of publication: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1043
Email: bhoai@umich.edu

Charles R. Johnson
Affiliation: Department of Mathematics, College of William & Mary, Williamsburg, Virginia 23185
Email: crjohnso@math.wm.edu

Ilya M. Spitkovsky
Affiliation: Department of Mathematics, College of William & Mary, Williamsburg, Virginia 23185
Email: ilya@math.wm.edu

DOI: 10.1090/S0002-9939-08-09104-1
PII: S 0002-9939(08)09104-1
Keywords: Semidefinite operators/matrices, spectral order, power dominance
Received by editor(s): November 30, 2006,
Received by editor(s) in revised form: January 3, 2007
Posted: February 14, 2008
Additional Notes: The work on this paper in the summer of 2006 was supported in part by the National Science Foundation Grant No. DMS-0353510
The third author (IMS) is also partially supported by the National Science Foundation Grant No. DMS-0456625.
Communicated by: Joseph A. Ball
Copyright of article: Copyright 2008, American Mathematical Society

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