On spectra of some completely positive maps

Let \(\sum _{i=1}^{\infty }A_iA_i^*\) ${\sum }_{i=1}^{\infty }{A}_i{A}_i^{\ast }$ and \(\sum _{i=1}^{\infty }A_i^*A_i\) ${\sum }_{i=1}^{\infty }{A}_i^{\ast }{A}_i$ converge in the strong operator topology. We study the map \(\Phi _{{\mathcal {A}}}\) ${\mathrm{\Phi }}_{\mathcal{A}}$ defined on the Banach space of all bounded linear operators \({\mathcal {B(H)}}\) $\mathcal{B}(\mathcal{H})$ by \(\Phi _{{\mathcal {A}}}(X)=\sum _{i=1}^{\infty }A_iXA_i^*\) ${\mathrm{\Phi }}_{\mathcal{A}}(X)={\sum }_{i=1}^{\infty }{A}_{i}X{A}_i^{\ast }$ and its restriction \(\Phi _{{\mathcal {A}}}|_{\mathcal {K(H})}\) ${\mathrm{\Phi }}_{\mathcal{A}}{|}_{\mathcal{K}(\mathcal{H})}$ to the Banach space of all compact operators \(\mathcal {K(H)}.\) $\mathcal{K}(\mathcal{H}).$ We first consider the relationship between the boundary eigenvalues of \(\Phi _{{\mathcal {A}}}|_{\mathcal {K(H})}\) ${\mathrm{\Phi }}_{\mathcal{A}}{|}_{\mathcal{K}(\mathcal{H})}$ and its fixed points. Also, we show that the spectra of \(\Phi _{{\mathcal {A}}}\) ${\mathrm{\Phi }}_{\mathcal{A}}$ and \(\Phi _{{\mathcal {A}}}|_{\mathcal {K(H})}\) ${\mathrm{\Phi }}_{\mathcal{A}}{|}_{\mathcal{K}(\mathcal{H})}$ are the same sets. In particular, the spectra of two completely positive maps involving the unilateral shift are described.