IAM   02674
INSTITUTO ARGENTINO DE MATEMATICA ALBERTO CALDERON
Unidad Ejecutora - UE
artículos
Título:
Weighted least squares solutions of the equation AXB - C = 0
Autor/es:
JUAN IGNACIO GIRIBET; ALEJANDRA MAESTRIPIERI; MAXIMILIANO CONTINO
Revista:
LINEAR ALGEBRA AND ITS APPLICATIONS
Editorial:
ELSEVIER SCIENCE INC
Referencias:
Lugar: Amsterdam; Año: 2017 vol. 518 p. 177 - 177
ISSN:
0024-3795
Resumen:
Let H be a Hilbert space, L(H) the algebra of bounded linear operators on H and W ∈ L(H) a positive operator such that W^1/2 is in the p-Schatten class, for some 1 ≤ p < ∞. Given A,B ∈ L(H) with closed range and C ∈ L(H), we study the following weighted approximation problem: analyze the existence ofmin{ ||AXB − C||p,W , X ∈L(H)}, (0.1)where ||X ||p,W = ||W^1/2 X ||p . We also study the related operator approximation problem: analyze the existence ofmin {(AXB − C)*W (AXB − C), X ∈L(H)}, (0.2)where the order is the one induced in L(H) by the cone of positive operators. In this paper we prove that the existence of the minimum of (0.2) is equivalent to the existence of a solution of the normal equation A*W (AXB − C) = 0. We also give sufficient conditions for the existence of the minimum.