Necessary Condition for Vector-Valued Model Spaces to be Invariant Under Conjugation

Abstract

The $S^{*}$-invariant subspaces of the Hardy-Hilbert space $H^{2}(E)$ (where $E$ is finite dimensional Hilbert space of dimension greater than 1) on the unit disc is well known. In this study, we examine that, if $\Omega$ is a conjugation on $E$, and $\Theta$ an inner function, then there exist model spaces which are not invariant for the conjugation $C_{\Omega}:L^{2}(E)\longrightarrow L^{2}(E)$. Under what necessary condition the model spaces is mapped onto itself is under consideration.