Let \(\sum _{i=1}^{\infty }A_iA_i^*\) and \(\sum _{i=1}^{\infty }A_i^*A_i\) converge in the strong operator topology. We study the map \(\Phi _{\mathcal {A}}\) defined on the Banach space of all bounded linear operators \({\mathcal {B(H)}}\) by \(\Phi _{\mathcal {A}}(X)=\sum _{i=1}^{\infty }A_iXA_i^*\) and its restriction \(\Phi _{\mathcal {A}}|_{\mathcal {S}_p\mathcal {(H)}}\) to the Banach space of all Schatten p-class operators \({\mathcal {S}_p\mathcal {(H)}}.\) We first consider the relationship between the spectra and the norms of \(\Phi _{\mathcal {A}}|_{\mathcal {S}_p\mathcal {(H)}}\) and \(\Phi ^\dag _{\mathcal {A}}|_{\mathcal {S}_p\mathcal {(H)}},\) where \(\Phi ^\dag _{\mathcal {A}}\) is the dual of \(\Phi _{\mathcal {A}}.\) Moreover, we present the structure and some equivalent conditions under which \(\Phi _{\mathcal {A}}|_{\mathcal {S}_p\mathcal {(H)}}\) is an isometry.
