Root lineals and characteristic functions of contractions

It is proved that $\operatorname{dim}\operatorname{Ker}(T-\lambda I)^n=\operatorname{dim}\operatorname{Ker}K_n(\lambda)$, where $T$ is a contraction of the Hilbert space,
$$ K_n(\lambda) =\begin{pmatrix} \theta^{*}(\bar\lambda)&0&\dots&0\\ \frac1{1!}\theta^{*(1)}(\bar\lambda)&\theta^{*}(\bar\lambda)&\dots&0\\ \frac1{(n-1)!}\theta^{*(n-1)}(\bar\lambda)&\frac1{(n-2)!}\theta^{*(n-2)}(\bar\lambda)&\dots& \theta^{*}(\bar\lambda) \end{pmatrix}, $$
$\theta$ – the characteristic function of the operator collegation generated by the contraction $T$.