Functional Analysis (10): Semigroups of Operators — Evolution Equations in Infinite Dimensions

Tue, 19 Oct 2021 09:00:00 +0000

The simplest interesting differential equation is $$u' = a u$$ , with $a \in \mathbb{R}$ . The solution $u(t) = e^{at} u_0$ is so familiar that it is easy to forget it is a piece of structure: the map $T(t) = e^{at}$ is a one-parameter family of operators on $\mathbb{R}$ satisfying $$T(0) = I$$ , $$T(t + s) = T(t) T(s)$$ , and continuity in $$t$$ . Replace $$a$$ with a self-adjoint matrix $$A$$ and you have $T(t) = e^{tA}$ , the matrix exponential, which solves the system $$u' = Au$$ . Replace $$A$$ with an unbounded operator on a Hilbert space — the Laplacian, the Schrödinger Hamiltonian, a Fokker-Planck operator — and you would like to do the same thing. But the matrix-exponential power series may not converge, the operator may not be defined on all of $$H$$ , and ordinary calculus stops working.

Semigroups on Chen Kai Blog

Functional Analysis (10): Semigroups of Operators — Evolution Equations in Infinite Dimensions