Norm inequality of \(AP+BQ\) for selfadjoint projections \(P\) and \(Q\) with \(PQ=0\) (Q2855188)

Let \(A\), \(B\) be bounded linear operators on a Hilbert space \(\mathcal L\) and \(P\), \(Q\) be selfadjoint projections on \(\mathcal L\) with \(PQ=0\). For a bounded linear operator \(X\) on \(\mathcal L\), define a Hankel type operator \(H_{X}:P\mathcal L\to Q\mathcal L\) by \(H_{X}=QX|_{P\mathcal L}\). The main results of the paper are as follows.NEWLINENEWLINE(1) Suppose that \(\alpha\), \(\beta\in\mathbb C\), \(A^{\ast}A=|\alpha|^{2}I\), \(B^{\ast}B=|\beta|^{2}I\). Then NEWLINE\[NEWLINE \| AP+BQ \|^{2} = \frac{|\alpha|^{2}+|\beta|^{2}}{2} + \sqrt { \| H_{B^{\ast}A}\|^{2} + \left( \frac{|\alpha|^{2}-|\beta|^{2}}{2} \right)^{2}. } NEWLINE\]NEWLINE (2) For any bounded operators \(A, B\), NEWLINE\[NEWLINE \max(\| AP\|^{2},\| BQ\|^{2}) \leq \| AP+BQ \|^{2} \leq \max(\| AP\|^{2},\| BQ\|^{2}) + \| H_{B^{\ast}A}\|. NEWLINE\]