(left) Shift Semigroup and operator norm

Question

I am reading some lecture notes on strongly continuous semigroups. I am having difficulty understanding an example:
$$X = BUC(\mathbb{R}) := \{f:\mathbb R \rightarrow \mathbb R : f \text{ is uniformly continuous and bounded}\} $$
with supremum norm $$\|f\|_{\infty} = \sup _{s \in \mathbb R}|f(s)|$$
$T(t)$ is the shift semigroup; $T(t)f(s) = f(t+s)$, which is indeed a strongly continuous semigroup.
Then it says $T(t)$ is not continuous for the operator norm.
How can it be strongly continuous semigroup on X but not continuous for the operator norm?
I mean it is obviously bounded; what am I missing here?

Answer 1

Strongly continuous means $$ \forall x: \quad \lim_{t\to0}\|T(t)x-x\|_X \to 0, $$ while continuity in operator norm means $\|T(t)-Id\|_{\mathcal L(X,X)}\to 0$, which is equivalent to $$ \lim_{t\to0} \sup_{x: \|x\|\le1} \|T(t)x-x\|_X \to 0, $$ in the second case the convergence has to be uniform with respect to $x$.

In your example, the first property is a consequence of uniform continuity of $x\in X$. To show that the second property is not fulfilled, it is enought to construct for each $\epsilon>0$ a function $x\in X$ such that $\|T(t)x-x\|>\epsilon$. This can be achieved using $x(s):=\max(-1,\min(ns,1))$ for sufficiently large $n$.