On spectra of some completely positive maps

Abstract

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.