In the paper Partial isometries and generalized inverses by Mbekhta, a bounded linear operator $T$ on a complex Banach space $X$ is defined to be a partial isometry if it is a contraction and it admits a generalized inverse that is also a contraction, where a generalized inverse $S$ of $T$ is a bounded linear operator on $X$ such that $STS=S$ and $TST=T$. There is also a remark after 4.3 that for $L^p$ spaces ($1\leq p\leq\infty$), all isometries are partial isometries in this sense.
In the case of Hilbert space, if $T$ is an isometry, then $T^*$ is a generalized inverse of $T$, and $T^*T=I$. For $L^p$ spaces ($1<p<\infty$), if $T$ is an isometry, can we always find a generalized inverse $S$ such that $ST=I$?