|
Another observation about operator compressions
Author(s):
Elizabeth
S.
Meckes;
Mark
W.
Meckes
Journal:
Proc. Amer. Math. Soc.
139
(2011),
1433-1439.
MSC (2010):
Primary 47A20, 60E15, 60B20
Posted:
September 2, 2010
MathSciNet review:
2748436
Retrieve article in:
PDF
Abstract |
References |
Similar articles |
Additional information
Abstract:
Let  be a self-adjoint operator on a finite dimensional Hilbert space. It is shown that the distribution of the eigenvalues of a compression of  to a subspace of a given dimension is almost the same for almost all subspaces. This is a coordinate-free analogue of a recent result of Chatterjee and Ledoux on principal submatrices. The proof is based on measure concentration and entropy techniques, and the result improves on some aspects of the result of Chatterjee and Ledoux.
References:
-
- 1.
- R. Bhatia.
Matrix Analysis, volume 169 of Graduate Texts in Mathematics.
Springer-Verlag, New York, 1997. MR 1477662 (98i:15003) - 2.
- S. Chatterjee.
Concentration of Haar measures, with an application to random matrices.
J. Funct. Anal., 245(2):379-389, 2007. MR 2309833 (2008a:60011) - 3.
- S. Chatterjee and M. Ledoux.
An observation about submatrices.
Elect. Comm. in Probab., 14:495-500, 2009. MR 2559099 - 4.
- R.M. Dudley.
The sizes of compact subsets of Hilbert space and continuity of Gaussian processes.
J. Functional Analysis, 1:290-330, 1967. MR 0220340 (36:3405) - 5.
- M. Gromov and V. D. Milman.
A topological application of the isoperimetric inequality.
Amer. J. Math., 105(4):843-854, 1983. MR 708367 (84k:28012) - 6.
- M. Ledoux.
The Concentration of Measure Phenomenon, volume 89 of Mathematical Surveys and Monographs.
American Mathematical Society, Providence, RI, 2001. MR 1849347 (2003k:28019) - 7.
- E. Meckes.
Approximation of projections of random vectors.
To appear in J. Theoret. Probab.; DOI 10.1007/s10959-010-0299-2. - 8.
- M. Talagrand.
Majorizing measures: the generic chaining.
Ann. Probab., 24(3):1049-1103, 1996. MR 1411488 (97k:60097) - 9.
- A. W. van der Vaart and J. A. Wellner.
Weak Convergence and Empirical Processes.
Springer Series in Statistics. Springer-Verlag, New York, 1996.
With applications to statistics. MR 1385671 (97g:60035) - 10.
- C. Villani.
Topics in Optimal Transportation, volume 58 of Graduate Studies in Mathematics.
American Mathematical Society, Providence, RI, 2003. MR 1964483 (2004e:90003)
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical
Society
with
MSC (2010):
47A20, 60E15, 60B20
Retrieve articles in all Journals with
MSC (2010):
47A20, 60E15, 60B20
Additional Information:
Elizabeth
S.
Meckes
Affiliation:
Department of Mathematics, Case Western Reserve University, Cleveland, Ohio 44106
Email:
elizabeth.meckes@case.edu
Mark
W.
Meckes
Affiliation:
Department of Mathematics, Case Western Reserve University, Cleveland, Ohio 44106
Email:
mark.meckes@case.edu
DOI:
10.1090/S0002-9939-2010-10671-8
PII:
S 0002-9939(2010)10671-8
Received by editor(s):
January 12, 2010
Received by editor(s) in revised form:
April 29, 2010
Posted:
September 2, 2010
Additional Notes:
The first author’s research was supported by an American Institute of Mathematics Five-Year Fellowship and NSF grant DMS-0852898.
The second author’s research was supported by NSF grant DMS-0902203.
Communicated by:
Marius Junge
Copyright of article:
Copyright
2010,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
