Find the dual operator $T^*$ where $T: L_p(\mathbb{R}) \to L_p(\mathbb{R})$ and $Tf(t) = f(t+c)$

Question

Assume some $c \in \mathbb{R}$ and that $1 \leq p < \infty$.

The dual operator is defined as $T^*:L_p(\mathbb{R})^* \to L_p(\mathbb{R})^*$ where $T^*\varphi(x)=\varphi(Tx)$

We also have $L_p^* = L_q$ where $q=\frac{p}{p-1}$ so actually we have $T^*:L_q(\mathbb{R}) \to L_q(\mathbb{R})$

Though, not sure how to proceed from here to actually finiding $T^*$?

Answer 1

Let $f \in L^{q}$. You want to find $h \in L^{p}$ such that $\int h(t)g(t)dt =\int f(t)g(t+c)dt$ for all $g \in L^{p}$. Make change of variable on RHS to write this as $\int h(t)g(t)dt =\int f(t-c)g(t)dt$. This hods for all $g \in L^{p}$ iff $h(t)=f(t-c)$ almost everywhere. Hence $h=T^{*}f$ is the function $f(t-c)$