Boundary values of a convergent sequence of J-contractive matrix-functions

D. Z. Arov ; L. A. Simakova

Matematičeskie zametki, Tome 19 (1976) no. 4, pp. 491-500

Résumé

In this note it is proved that if $W_n(z)$ are $J$-contractive matrix-functions which are meromorphic in the disk $|z|<1$ ($J-W_n^*(z)JW_n(z)\ge0$, $J^*=J$, $J^2=I$), $W_n(z)\to W(z)$, as $n\to\infty$, $$ W^*(z)JW(z)\le W_n^*(z)JW_n(z) $$ and $$ \det W(z)\not\equiv0, $$ then there exists a subsequence $W_{n_k}(z)$ whose boundary values $$ W^*_{n_k}(\zeta)JW_{n_k}(\zeta)\to W^*(\zeta)JW(\zeta)\quad (\text{a. e. }|\zeta|=1). $$ It follows from this result that every convergent Blaschke–Potapov product has $J$-unitary boundary values.