On spectra of some completely positive maps

Let ∑ i = 1 ∞ A i A i ∗ and ∑ i = 1 ∞ A i ∗ A i converge in the strong operator topology. We study the map Φ A defined on the Banach space of all bounded linear operators B ( H ) by Φ A ( X ) = ∑ i = 1 ∞ A i X A i ∗ and its restriction Φ A | K ( H ) to the Banach space of all compact operators K ( H ) . We first consider the relationship between the boundary eigenvalues of Φ A | K ( H ) and its fixed points. Also, we show that the spectra of Φ A and Φ A | K ( H ) are the same sets. In particular, the spectra of two completely positive maps involving the unilateral shift are described.