Two-dimensional reductions of the cone of positive diagonal operators in \(\ell^{2}\). (Q1405052)

This short paper is devoted to the reductions of the cones of positive operators acting on the Hilbert space \(\ell^{2}\). Let \(\mathcal B\) denote the set of all bounded linear operators on \(\ell^{2}\). The author proves that for any two-dimensional orthoprojection \(q\) in the space \(\ell^{2}\), one has \(q{\mathcal D}^{+}q\neq q{\mathcal B}^{+}q\), where \({\mathcal B}^{+}\) and \({\mathcal D}^{+}\) denote the cone of all positive operators in \(\mathcal B\) and the cone of all positive diagonal operators in \(\mathcal B\), respectively.