Geodesic
Polar decomposition under perturbations of the scalar product
The electronic journal of linear algebra, Tome 7 (2000), pp. 21-29.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: Let A be a unital C $\Lambda $-algebra with involution $\Lambda $represented in a Hilbert space H, G the group of invertible elements of A, U the unitary group of A, G s the set of invertible selfadjoint elements of A, Q = f" 2 G : " 2 = 1g the space of reflections and P = Q " U. For any positive a 2 G consider the a-unitary group U a = fg 2 G : a $\Gamma 1$ g $\Lambda a = g \Gamma 1$ g, i.e., the elements which are unitary with respect to the scalar product h; ji a = ha; ji for ; j 2 H. If ss denotes the map that assigns to each invertible element its unitary part in the polar decomposition, it is shown that the restriction ssj U a : U a ! U is a diffeomorphism, that $ss(U a " Q) = P$ , and that $ss(U a " G s ) = U$ a " G s = fu 2 G : u = u $\Lambda = u \Gamma 1$ and au = uag:
Classification : 47A30, 47B15
Keywords: polar decomposition, C $\Lambda $-algebras, positive operators, projections 
@article{ELA_2000__7__a12,
     author = {Corach, Gustavo and Maestripieri, Alejandra and Stojanoff, Demetrio},
     title = {Polar decomposition under perturbations of the scalar product},
     journal = {The electronic journal of linear algebra},
     pages = {21--29},
     publisher = {mathdoc},
     volume = {7},
     year = {2000},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ELA_2000__7__a12/}
}
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Corach, Gustavo; Maestripieri, Alejandra; Stojanoff, Demetrio. Polar decomposition under perturbations of the scalar product. The electronic journal of linear algebra, Tome 7 (2000), pp. 21-29. http://geodesic.mathdoc.fr/item/ELA_2000__7__a12/