Semigroups of Linear Operators

Our goal is to define exponentials of linear operators. We will try to construct etA as a linear operator, where A : D(A)→ X is a general linear operator, not necessarily bounded. Notationally, it seems like we are looking for a solution to μ̇(t) = Aμ(t), μ(0) = μ0, and we would like to write μ(t) = eμ0. It turns out that this will hold once we make sense of the terms. How can we construct etA when A is a finite matrix? The most obvious way is to write down the power series: ∞ n=0 1 n! (tA) n. This series is absolutely convergent for every A and t ∈ R. In fact, this method works for A ∈ L (X ; X ), even if X is infinite dimensional.