Let $H$ be a Hilbert space and $H'\le H$ a subspace as Hilbert spaces (I mean, the inner product in $H'$ is the same inner product of $H$ restricted to $H'$).
If we take $f:H\to H$ an automorphism of Hilbert spaces which fixes $H'$ pointwise, notice that if $f(a)=b$ then $\|a\|=\|b\|$ and $dist(a,H')=dist(b,H')$. Is the converse also true? (I mean, if $a,b\in H$ satisfy $\|a\|=\|b\|$ and $dist(a,H')=dist(b,H')$, is there an automorphism $f:H\to H$ which fixes $H'$ pointwise and $f(a)=b$?).