If $AA^*$ is an isomorphism does this imply that $A^*A$ is an isomorphism?

Question

Let $H_1$ and $H_2$ be two Hilbert spaces and let $A:H_1\to H_2$ be linear such that $AA^*$ is an isomorphism. Does this imply that $A^*A$ is an isomorphism?

Answer 1

No: suppose that $H_1=\mathbb{R}^2$ and $H_2=\mathbb{R}$, and define $A$ by $\begin{bmatrix}x\\y\end{bmatrix}\mapsto x$.

Then $A^*$ maps $x$ to $\begin{bmatrix}x\\0\end{bmatrix}$, so $AA^*$ is the identity on $\mathbb{R}$, but $A$ is not an isomorphism.

Answer 2

Consider $Vect(e_i,i\geq 0)$, $A(e_i)=e_{i-1}, i\geq 1, A^*(e_0)=0$. $\langle A(e_i),e_j\rangle=\langle e_{i-1},e_j\rangle=\langle e_i,A^*(e_j)\rangle$ implies that $A^*(e_{i})=e_{i+1}, i\geq 1, A^*(e_0)=0$ $AA^*=Id$.