Given a Hilbert Space $(H,\langle,\rangle)$, $x,y\in H$ and $D\subset H$ a subspace of $H$ (I mean, the operators $+$, $\cdot$ and $\langle,\rangle$ in D are the restrictions of the respective ones in $H$), the fact that there exists an isometry $f:H\to H$ such that $f(x)=y$ and $f(z)=z$ for every $z\in D$ is characterized by having $\|x\|=\|y\|$ and $Proy_D(x)=Proy_D (y)$.
Is there any way to get a similar characterization in general Banach Spaces?
Thanks.