Consider a linear operator $V$ between two Hilbert spaces $H_A$ and $H_B$, we say that $V:H_A\rightarrow H_B$ is an isometry if $$ V^*V=\mathbb{1}_A$$
if
$V$ is an isometry, does it always hold that $VV^*=\mathbb{1}_B$?
In other words, is the adjoint on an isometry an isometry?
If not, does it in finite dimension? If not even in finite dimension, does anyone have a finite dimensional counterexample?
Best Answer
It will hold in finite dimensional spaces if $\dim(H_A) = \dim(H_B)$. It will not, however, hold more generally.
If we take $H_A = \Bbb C^2$ and $H_B = \Bbb C^3$, then the map $$ V(x,y) = (x,y,0) $$ is an isometry. Its adjoint $$ V^*(x,y,z) = (x,y) $$ is not an isometry.
For an infinite dimensional example with $H_A = H_B$, we can take the right shift operator over $\ell^2$.