Fixed points of completely positive maps and their dual maps

Abstract

AbstractLet $\mathcal {A} \subset{\mathcal {B}}(\mathcal {H})$ A ⊂ B ( H ) be a row contraction and $\Phi _{\mathcal {A}}$ Φ A determined by $\mathcal {A}$ A be a completely positive map on ${\mathcal {B}}(\mathcal {H})$ B ( H ) . In this paper, we mainly consider fixed points of $\Phi _{\mathcal {A}}$ Φ A and its dual map $\Phi _{\mathcal {A}}^{\dagger}$ Φ A † . It is given that $\Phi _{\mathcal {A}}(X)\leq X $ Φ A ( X ) ≤ X (or $\Phi _{\mathcal {A}}(X)\geq X $ Φ A ( X ) ≥ X ) implies $\Phi _{\mathcal {A}}(X)= X$ Φ A ( X ) = X and $\Phi _{\mathcal {A}}^{\dagger}(X)= X$ Φ A † ( X ) = X when $X\in {\mathcal {B}}(\mathcal {H})$ X ∈ B ( H ) is a compact operator. Some necessary conditions of $\Phi _{\mathcal {A}}(X)= X$ Φ A ( X ) = X and $\Phi _{\mathcal {A}}^{\dagger}(X)= X$ Φ A † ( X ) = X are given.