Operator Cauchy equation and one-parameter semigroups

Remarks

Conditions 2 and 3 make $T$ continuous in operator norm on $[0, \infty)$. Let $V(t) = \int_0^t T(s) \, ds$; since $V(t)/t \to I$ in norm, $V(t_0)$ is invertible for a small $t_0 > 0$. The functional equation gives $(T(h) - I) V(t_0) = V(t_0 + h) - V(h) - V(t_0)$, and dividing by $h$ shows that $T$ is differentiable at $0$ with derivative $A = (T(t_0) - I) V(t_0)^{-1}$. Then $T'(t) = A T(t)$, and uniqueness for this linear equation gives $T(t) = e^{tA}$. The operator $A$ is the generator of the semigroup. For $X = \R^n$ this says that the continuous solutions $F \colon [0, \infty) \to \R^{n \times n}$ of $F(s + t) = F(s) F(t)$ with $F(0) = I$ are exactly $F(t) = e^{tA}$. Condition 3 cannot be weakened to strong continuity. The left shifts $(T(t)f)(s) = f(s + t)$ on $L^2(\R)$ form a $C_0$-semigroup with the unbounded generator $f \mapsto f'$, so they are not of the form $e^{tA}$ with $A$ bounded. Condition 1 is needed: for a projection $P \ne I$, the constant map $T(t) = P$ satisfies condition 2 and is norm continuous, but $T(0) = P \ne e^{0A}$.

Going deeper

For a student

The theorem identifies the norm-continuous one-parameter semigroups with the bounded operators $A$. Many evolution equations, such as the heat equation, give semigroups that are only strongly continuous, and then the generator is an unbounded operator. For those semigroups the Hille–Yosida theorem takes the place of this result.