Semigroup of Bounded Linear Operators Uniformly Continuous iff Continuous as Map from Non-Negative Reals to Bounded Linear Operators

Theorem
Let $\GF \in \set {\R, \C}$.

Let $X$ be a Banach space over $\GF$.

Let $\family {\map T t}_{t \ge 0}$ be a semigroup of bounded linear operators.

Let $\struct {\map B X, \norm {\, \cdot \,}_{\map B X} }$ be the space of bounded linear transformations equipped with the canonical norm.

Then $\family {\map T t}_{t \ge 0}$ is uniformly continuous :


 * the mapping $T : \hointr 0 \infty \to \map B X$ is continuous.

Necessary Condition
If $T$ is continuous, then in particular it is continuous at $0$.

Since $\map T 0 = I$, we therefore have:


 * $\ds \lim_{t \mathop \to 0^+} \norm {\map T t - I}_{\map B X} = 0$

So $\family {\map T t}_{t \ge 0}$ is uniformly continuous.

Sufficient Condition
Suppose that $\family {\map T t}_{t \ge 0}$ is uniformly continuous. We have that:


 * $\ds \lim_{t \mathop \to 0^+} \norm {\map T t - I}_{\map B X} = 0$

so $T$ is continuous at $0$.

Let $t > 0$ and $h > 0$.

Then, we have:

so:


 * $\ds \lim_{h \to 0^+} \norm {\map T {t + h} - \map T t}_{\map B X} = 0$

Now let $-t < h < 0$.

Then $h + t \in \closedint 0 t$ for all such $h$, so there exists $M > 0$:


 * $\norm {\map T {t + h} }_{\map B X} \le M$

for all $-t < h < 0$.

Then we have:

So we have:


 * $\ds \lim_{h \to 0^-} \norm {\map T {t + h} - \map T t}_{\map B X} = 0$

and hence:


 * $\ds \lim_{h \to 0} \norm {\map T {t + h} - \map T t}_{\map B X} = 0$

for $t > 0$.

So $T : \hointr 0 \infty \to \map B X$ is continuous.