Partially isometric approximation of positive operators (Q909962)

Let A be a positive operator on a Hilbert space and consider the quantity \(\| A-U\|_ p,\) where U runs through the set of all unitaries such that A-U is contained in the Schatten class \(\ell_ p\). \textit{J. G. Aiken}, \textit{J. A. Erdős} and \textit{J. A. Goldstein} proved [Ill. J. Math. 24, 61-72 (1980; Zbl 0404.47014)] that for \(1\leq p<\infty\) \(\| A-U\|_ p\) is minimized when \(U=I\) and, in the finite dimensional case, maximized when \(U=-I.\) The present paper deals with the same problems in the case where U runs, respectively, through the isometries and the partial isometries of the underlying Hilbert space. The case of isometric approximation turns out to be exactly alike to the unitary approximation. In fact, it is proved that if A-U is in \(\ell_ p\) and U is an isometry, then U is unitary. Partial isometric approximation turns out to be more difficult to deal with. One of the main results goes as follows: Let A be a positive operator and consider those partial isometries U for which \(A-U\in \ell_ p\), where \(1<p<\infty\). If the map \(U\mapsto \| A-U\|^ p_ p\) attains a global minimum, then the underlying Hilbert space has a basis \(\{\phi_ n\}\) consisting of eigenvalues of A, and \(\| A-U\|_ p\geq \| A-E_{}\|_ p,\) where \(E_{}\) is the projection into the closed subspace generated by those \(\phi_ n\) pertaining to eigenvalues \(\geq\).