Let
a1142a947d44e9d01c2d3b9" title="Click to view the MathML source">H be a Hil
bert space,
9eef978474e7c1ad8117c2518b8311b9" title="Click to view the MathML source">L(H) the algebra of bounded linear operators on
a1142a947d44e9d01c2d3b9" title="Click to view the MathML source">H and
W∈L(H) a positive operator such that
e5b10d04111fcb66e2b8a" title="Click to view the MathML source">W1/2 is in the p-Schatten class, for some
a185e2121440427cfab" title="Click to view the MathML source">1≤p<∞. Given
9bce5c62697165f30" title="Click to view the MathML source">A∈L(H) with closed range and
9e53dd24a664e9fd65a97a" title="Click to view the MathML source">B∈L(H), we study the following weighted approximation problem: analyze the existence of
where
‖X‖p,W=‖W1/2X‖p. In this paper we prove that the existence of this minimum is equivalent to a compatibility condition
between
R(B) and
R(A) involving the weight
W, and we characterize the operators which minimize this problem as
W-inverses of
A in
R(B).