On spectra of some completely positive maps

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 \(\Phi _{{\mathcal {A}}}|_{\mathcal {K(H})}\) to the Banach space of all compact operators \(\mathcal {K(H)}.\) We first consider the relationship between the boundary eigenvalues of \(\Phi _{{\mathcal {A}}}|_{\mathcal {K(H})}\) and its fixed points. Also, we show that the spectra of \(\Phi _{{\mathcal {A}}}\) and \(\Phi _{{\mathcal {A}}}|_{\mathcal {K(H})}\) are the same sets. In particular, the spectra of two completely positive maps involving the unilateral shift are described.