Remark on the positive contracting approximation (Q1300885)

Let \(\mathcal I\) be a proper bilateral ideal of the algebra \({\mathcal B}({\mathcal H})\) of all bounded linear operators on some complex Hilbert space \(\mathcal H\). Let \(\|\cdot\|_{\mathcal I}\) denote the symmetric norm of \(\mathcal I\) and \(A\), \(P\) be bounded linear operators on \(\mathcal H\) such that \(A-P\in\mathcal I\). Then \(P\) is called a positive contracting approximation of \(A\) (with respect to \(\|\cdot\|_{\mathcal I}\)) if \(P\) is a positive contraction such that \[ \|A-P\|_{\mathcal I}=\inf\{\|A-R\|_{\mathcal I}\mid R\in{\mathcal B}({\mathcal H}), A-R\in{\mathcal I}\}. \] In the paper under review, a simple formula is given for the positive contracting approximation of \(A\) when \(A\) is a normal operator. If moreover the ideal \(\mathcal I\) is one of the Schatten-von Neumann classes \({\mathcal C}_p\) with \(1<p<\infty\), then the approximant is unique.