Let $f:\mathbb{R}\to \mathbb{C}$ be a function in $L^p(\mathbb{R})$, with $1\leq p\leq \infty$. For $a\in \mathbb{R}$, define the translation operator $\tau_a$ by $\tau_a f(x)=f(x-a)$.
Let $$ Tf =f+i\frac{\tau_a f -\tau_{-a}f}{2}, $$ where $i^2=-1$. Clearly $T$ is a bounded linear operator in $L^p$ (in fact, it should be in any rearrangement invariant space).
The question is the following: Does $T$ possess an inverse, and if the answer is positive, is there a closed form for $T^{-1}$? I have tried finding such an inverse explicitly by considering $$ T^{-1}f=f+i \frac{\pm \tau_a f \pm\tau_{-a}f}{2} $$ without any success.