Reducing Subspaces of de Branges-Rovnyak Spaces

For $b\in H^\infty_1$, the closed unit ball of $H^\infty$, the de Branges-Rovnyak spaces $\mathcal{H}(b)$ is a Hilbert space contractively contained in the Hardy space $H^2$ that is invariant by the backward shift operator $S^*$. We consider the reducing subspaces of the operator $S^{*2}|_{\mathcal{H}(b)}$. When $b$ is an inner function, $S^{*2}|_{\mathcal{H}(b)}$ is a truncated Toepltiz operator and its reducibility was characterized by Douglas and Foias using model theory. We use another approach to extend their result to the case where $b$ is extreme. We prove that if $b$ is extreme but not inner, then $S^{*2}|_{\mathcal{H}(b)}$ is reducible if and only if $b$ is even or odd, and describe the structure of reducing subspaces.