Strongly continuous shifted orbits in finite Bochner norms

Fix $h\in H$ and $r\in [0,T]$. Uniform approximation of the continuous orbit on $[0,T]$ by finitely many sampled values on increasingly fine interval partitions, followed by the measurable cutoff, proves strong measurability of $R_{r}h$ by [F4]. Unitarity gives $\mathrm{\parallel }{R}_{r}h{\mathrm{\parallel }}_{B_1}=(T-r)\mathrm{\parallel }h{\mathrm{\parallel }}_H\le T\mathrm{\parallel }h{\mathrm{\parallel }}_H$. If $r_n\in [0,T]$ tends to $r$, then $R_{r_n}h(t)\to R_{r}h(t)$ in $H$ for every $t\in (0,T)\setminus \{r\}$ by orbit continuity and eventual agreement of the cutoffs. Each row is bounded in norm by the integrable constant $\mathrm{\parallel }h{\mathrm{\parallel }}_H$. Thus [F3] gives $\mathrm{\parallel }{R}_{r_n}h-R_{r}h{\mathrm{\parallel }}_{B_1}\to 0$, also for $r=0,T$.

Write $A_h$ for the zero extension to $\mathbb{R}$ of a strongly measurable representative of the given $Y$-valued orbit. For any real $a$, translated simple approximants converge outside the translated exceptional null set, so [F4] and [F5] prove strong measurability of $A_h(\cdot -a)$ and independence from the chosen representative. For a nonnegative scalar simple function $s$, translating its level sets preserves each measure, whence ${\int }_{\mathbb{R}}s(t-a)dt={\int }_{\mathbb{R}}s(t)dt$. Translation bijects the simple minorants of a nonnegative measurable $g$ with those of $g(\cdot -a)$, so taking suprema proves the same identity for $g$ by [F5]. Apply it to $g(t)=\mathrm{\parallel }{A}_h(t){\mathrm{\parallel }}_Y^p$: whole-line translation preserves its Bochner $L^p$ norm. Since $0\le r\le T$, for $0<t<T$ one has $0<t-r<T$ exactly when $r<t<T$. Consequently the specified $B_2$ row is $A_h(\cdot -r){\mathrm{\mid }}_{(0,T)}$; restriction decreases the norm by the simple-minorant definition in [F5], so $\mathrm{\parallel }{R}_{r}h{\mathrm{\parallel }}_{B_2}\le \mathrm{\parallel }{A}_h{\mathrm{\parallel }}_{L^p(\mathbb{R};Y)}\le C_T\mathrm{\parallel }h{\mathrm{\parallel }}_H$.

Regard $A_h$ as a function on $(-2T,3T)$ and recenter this interval to $(0,5T)$. The local translation theorem [F2], applied after realification, gives $\mathrm{\parallel }{A}_h(\cdot -\delta )-A_h{\mathrm{\parallel }}_{L^p([-T,T];Y)}\to 0$ as $\delta \to 0$, since $[-T,T]$ is compactly contained in $(-2T,3T)$. For $r_n,r\in [0,T]$ and ${\delta }_n=r_n-r$, the change of variable $u=t-r$, justified by step 1.2, gives $\mathrm{\parallel }{R}_{r_n}h-R_{r}h{\mathrm{\parallel }}_{B_2}^p={\int }_{-r}^{T-r}\mathrm{\parallel }{A}_h(u-{\delta }_n)-A_h(u){\mathrm{\parallel }}_Y^{p}du$. This is bounded by the corresponding integral over $[-T,T]$ and tends to zero. This proves parameter continuity in $B_2$ without requiring pointwise $Y$ values at exceptional times.

By [F4], $H$ and $Y$ are complete; apply the real Bochner completeness theorem in [F2] to their realifications to obtain completeness of $B_1$ and $B_2$ in the unchanged metrics. Apply the interval-average assertion of [F2] with exponent one and value space the realification of $H$: the resulting interval-step functions converge to any prescribed $f\in L^1(0,T;H)$, proving density. For $0<r<T$ and $h\mathrm{\ne }0$, the left limit of the row at $r$ is $0$ and the right limit is $W(0)h=h$. A continuous representative agreeing almost everywhere would agree with the continuous row on each of $(0,r)$ and $(r,T)$, because a nonzero continuous difference persists on an interval of positive measure. Its two limits at $r$ would then disagree. This proves the claimed failure of continuous representability and completes the stated claims.