Reducing Subspaces of de Branges-Rovnyak Spaces

DOI10.1016/J.JMAA.2019.04.071arXiv1810.01058MaRDI QIDQ6307554

Cheng Chu

Publication date: 2 October 2018

Abstract: For $bin{H}^{i}nft{y}_1$, the closed unit ball of $H^{i}nfty$, the de Branges-Rovnyak spaces $mathcalH(b)$ 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 $S^{*2}{|}_{mathcalH(b)}$. When $b$ is an inner function, $S^{*2}{|}_{mathcalH(b)}$ 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 $S^{*2}{|}_{mathcalH(b)}$ is reducible if and only if $b$ is even or odd, and describe the structure of reducing subspaces.

Mathematics Subject Classification ID

Operator theory (47-XX) Functional analysis (46-XX)

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