Operators which remain convergent when multiplied by certain Hermitian operators (Q1965275)

Using the natural order \(A\geq B\) for Hermitian operators \(A\) and \(B\) when \(A-B\geq 0\) (i.e. \(A-B\) is positive semidefinite) and the order \(A\gg B\) when \(A\geq B\) and \(A-B\) is invertible, the author characterizes those Hermitian operators \(A\) for which \(MA\) has the spectral radius less than 1 (i.e. \(MA<1\)) for all \(M\in{\mathcal M}\) when \({\mathcal M}=[-I,I]\), \({\mathcal M}=[0,I]\) or \({\mathcal M}=(0,I]:= \{M\in[0,I]: M\geq 0\}\).