Geodesic
Some Convexity Results for the Cartan Decomposition
Canadian journal of mathematics, Tome 55 (2003) no. 5, pp. 1000-1018

Voir la notice de l'article provenant de la source Cambridge University Press

In this paper, we consider the set $\text{S}=a\left( {{e}^{X}}K{{e}^{Y}} \right)$ where $a\left( g \right)$ is the abelian part in the Cartan decomposition of $g$ . This is exactly the support of the measure intervening in the product formula for the spherical functions on symmetric spaces of noncompact type. We give a simple description of that support in the case of $\text{SL}\left( 3,\,\mathbf{F} \right)\,\text{where}\,\mathbf{F}\,=\,\mathbf{R},\,\mathbf{C}\,\text{or}\,\mathbf{H}$ . In particular, we show that $\text{S}$ is convex.We also give an application of our result to the description of singular values of a product of two arbitrary matrices with prescribed singular values.
DOI : 10.4153/CJM-2003-040-x
Mots-clés : 43A90, 53C35, 15A18, convexity theorems, Cartan decomposition, spherical functions, product formula, semisimple Lie groups, singular values 
Graczyk, P.; Sawyer, P. Some Convexity Results for the Cartan Decomposition. Canadian journal of mathematics, Tome 55 (2003) no. 5, pp. 1000-1018. doi: 10.4153/CJM-2003-040-x
@article{10_4153_CJM_2003_040_x,
     author = {Graczyk, P. and Sawyer, P.},
     title = {Some {Convexity} {Results} for the {Cartan} {Decomposition}},
     journal = {Canadian journal of mathematics},
     pages = {1000--1018},
     year = {2003},
     volume = {55},
     number = {5},
     doi = {10.4153/CJM-2003-040-x},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-2003-040-x/}
}
TY  - JOUR
AU  - Graczyk, P.
AU  - Sawyer, P.
TI  - Some Convexity Results for the Cartan Decomposition
JO  - Canadian journal of mathematics
PY  - 2003
SP  - 1000
EP  - 1018
VL  - 55
IS  - 5
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-2003-040-x/
DO  - 10.4153/CJM-2003-040-x
ID  - 10_4153_CJM_2003_040_x
ER  - 
%0 Journal Article
%A Graczyk, P.
%A Sawyer, P.
%T Some Convexity Results for the Cartan Decomposition
%J Canadian journal of mathematics
%D 2003
%P 1000-1018
%V 55
%N 5
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-2003-040-x/
%R 10.4153/CJM-2003-040-x
%F 10_4153_CJM_2003_040_x

[1] [1] Flensted-Jensen, M. and Koornwinder, T., The convolution structure for Jacobi expansions. Ark. Mat. 10(1973), 2–5–262. Google Scholar

[2] [2] Graczyk, P. and Sawyer, P., The product formula for the spherical functions on symmetric spaces in the complex case. Pacific J. Math. (2000), to appear. Google Scholar

[3] [3] Graczyk, P. and Sawyer, P., The product formula for the spherical functions on symmetric spaces of noncompact type. J. Lie Theory, to appear. Google Scholar

[4] [4] Hba, A., Analyse harmonique sur SL(3;) . C. R. Acad. Sci. Paris Série I 305(1987), 77–80. Google Scholar

[5] [5] Helgason, S., Differential Geometry, Lie Groups and Symmetric spaces. Academic Press, New York, 1978. Google Scholar

[6] [6] Helgason, S., Group and Geometric Analysis. Academic Press, New York, 1984. Google Scholar

[7] [7] Koornwinder, T., Jacobi polynomials, II. An analytic proof of the product formula. SIAM J. Math. Anal. (1) 5(1974), 125–137. Google Scholar

[8] [8] Markus, A. A., Eigenvalues and singular values of the sum and product of linear operators. Uspehi Mat. Nauk 19(1934), 93–123; RussianMath. Surveys (1964), 91–119. Google Scholar

[9] [9] Sawyer, P., The heat equation on spaces of positive definite matrices. Canad. J. Math. (3) 44(1992), 624–651. Google Scholar

[10] [10] Wigner, P., On a generalization of Euler's angles. In: Group theory and its applications, (ed., Loebl, Ernest M.), Academic Press, New York, London, 1968. Google Scholar

Cité par Sources :