show that for any complex number $\lambda$ such that $ |\lambda | < 1$ there exists a non-zero $x$ in $H$ such that $A^{\dagger}x = \lambda x$

Question

let $H$ be a complex Hilbert space, and $A$ a non-surjective isometric linear operator on $H$, $e_0 \in H$ a unit vector that is orthogonal to the range of $A$, $e_n = A^ne_0$ and $A^{\dagger}$ denotes the adjoint.

Question : show that for any complex number $\lambda$ such that $ |\lambda | < 1$ there exists a non-zero $x$ in $H$ such that $A^{\dagger}x = \lambda x$

I have shown the following, they might be useful :

• $\{e_n\}_{n \geq 0}$ is an orthonormal set
• $A^{\dagger}e_0 = 0$
• $A^{\dagger}e_n = e_{n-1}, \,\,\, \forall n \geq 1$