Embeddability of real and positive matrices

It is a well-known problem in probability theory whether a Markov matrix is embeddable into a Markov semigroup. Even today it is an active field of research. We consider a related problem: Given a (finite or infinite) matrix \(T\), is it embeddable into a real/positive \(C_0\)-semigroup, i.e., is there a real/positive \(C_0\)-semigroup \((T(t))_{t\geq 0}\) such that \(T(1)=T\)?

We will give necessary and sufficient conditions for embeddability of a real matrix into a real \(C_0\)-semigroup. Moreover, we will see that real-embeddability is typical for real contractions on \(l^2\).

In the case that \(T\) is positive we will present necessary conditions for embeddability of \(T\) into a positive \(C_0\) semigroup. In addition, we will give a full description of embeddability for positive \(2\times 2\) matrices.

The talk is based on joint work with Tanja Eisner.