Positive groups on \(\mathcal H^n\) are completely positive
For \(K=\mathbb R\) or \(K=\mathbb C\), let \(H^{n} \subset K^{n \times n}\) denote the real space of \(n \times n\) real or complex Hermitian matrices, and \(H_{+}^{n} \subset H^{n}\) the set of positive semidefinite matrices. A linear operator \(T: H^{n} \rightarrow H^{n}\) is said to be (i) positive if \(T(H_{+}^{n}) \subset H_{+}^{n},\) (ii) a Lyapunov operator if there is an \(A \in K^{n \times n}\) such that for all \(X \in H^{n}\), \(T(X)=AX+XA^{\star}\), where \(A^{\star}\) is the conjugate transpose of matrix \(A\). In this paper, the author proves that a linear operator \(T: H^{n} \rightarrow H^{n}\) is a generator of a positive group if and only if \(T\) is a Lyapunov operator. At the end of the paper, he asks the following question: Is it true that every exponentially positive operator on \(H^{n}\) can be decomposed in the sum of a Lyapunov operator and a positive operator?
scientific article; zbMATH DE number 2126346