Root lineals and characteristic functions of contractions

M. S. Brodskii ; Ya. S. Shvartsman

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part V, Tome 47 (1974), pp. 155-158

Résumé

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, $$ K_n(\lambda) =\begin{pmatrix} \theta^{*}(\bar\lambda)0\dots0\\ \frac1{1!}\theta^{*(1)}(\bar\lambda)\theta^{*}(\bar\lambda)\dots0\\ \frac1{(n-1)!}\theta^{*(n-1)}(\bar\lambda)\frac1{(n-2)!}\theta^{*(n-2)}(\bar\lambda)\dots \theta^{*}(\bar\lambda) \end{pmatrix}, $$ $\theta$ – the characteristic function of the operator collegation generated by the contraction $T$.