How to prove the following result involving $ TT^*$?

Question

If $T$ is a bounded linear operator between Hilbert spaces. Then how to prove that $$\left(\sum_{j=1}^{k} a_j(T^*T)^{j-1}\right)TT^*y = 0$$ if and only if $TT^*y$ is sum of at most $k-1$ eigen vectors of $TT^*$. Here $a_i$ are real numbers and $T^*$ is adjoint of $T$.