On log-subharmonicity of singular values of matrices

Bernard Aupetit

Studia Mathematica (1997)

  • Volume: 122, Issue: 2, page 195-200
  • ISSN: 0039-3223

Abstract

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Let F be an analytic function from an open subset Ω of the complex plane into the algebra of n×n matrices. Denoting by s 1 , . . . , s n the decreasing sequence of singular values of a matrix, we prove that the functions l o g s 1 ( F ( λ ) ) + . . . + l o g s k ( F ( λ ) ) and l o g + s 1 ( F ( λ ) ) + . . . + l o g + s k ( F ( λ ) ) are subharmonic on Ω for 1 ≤ k ≤ n.

How to cite

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Aupetit, Bernard. "On log-subharmonicity of singular values of matrices." Studia Mathematica 122.2 (1997): 195-200. <http://eudml.org/doc/216370>.

@article{Aupetit1997,
abstract = {Let F be an analytic function from an open subset Ω of the complex plane into the algebra of n×n matrices. Denoting by $s_1,...,s_n$ the decreasing sequence of singular values of a matrix, we prove that the functions $log s_\{1\}(F(λ)) + ... + log s_\{k\}(F(λ))$ and $log^\{+\}s_\{1\}(F(λ)) + ... + log^\{+\}s_\{k\}(F(λ))$ are subharmonic on Ω for 1 ≤ k ≤ n.},
author = {Aupetit, Bernard},
journal = {Studia Mathematica},
keywords = {Log-subharmonicity; singular values},
language = {eng},
number = {2},
pages = {195-200},
title = {On log-subharmonicity of singular values of matrices},
url = {http://eudml.org/doc/216370},
volume = {122},
year = {1997},
}

TY - JOUR
AU - Aupetit, Bernard
TI - On log-subharmonicity of singular values of matrices
JO - Studia Mathematica
PY - 1997
VL - 122
IS - 2
SP - 195
EP - 200
AB - Let F be an analytic function from an open subset Ω of the complex plane into the algebra of n×n matrices. Denoting by $s_1,...,s_n$ the decreasing sequence of singular values of a matrix, we prove that the functions $log s_{1}(F(λ)) + ... + log s_{k}(F(λ))$ and $log^{+}s_{1}(F(λ)) + ... + log^{+}s_{k}(F(λ))$ are subharmonic on Ω for 1 ≤ k ≤ n.
LA - eng
KW - Log-subharmonicity; singular values
UR - http://eudml.org/doc/216370
ER -

References

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  1. [1] B. Aupetit, A Primer on Spectral Theory, Springer, New York, 1991. 
  2. [2] B. Aupetit et A. Iyamuremye, Sous-harmonicité de la partie de Riesz du spectre d'un opérateur, Ann. Sci. Math. Québec 12 (1988), 171-177. 
  3. [3] H. König, Eigenvalue Distribution of Compact Operators, Birkhäuser, Basel, 1986. 
  4. [4] O. Nevanlinna, A characteristic function for matrix valued meromorphic functions, Helsinki University of Technology, Institute of Mathematics Research Report A 355, 1995. Zbl0869.30024
  5. [5] O. Nevanlinna, Meromorphic resolvents and power bounded operators, Helsinki University of Technology, Institute of Mathematics Research Report A 358, 1996. Zbl0865.65034

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