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Journal of Operator Theory

Volume 89, Issue 1, Winter 2023  pp. 105-123.

Higher order isometric shift operator on the de Branges--Rovnyak space

Authors:  Caixing Gu (1), Shuaibing Luo (2)
Author institution:(1) Department of Mathematics, California Polytechnic State University, San Luis Obispo, CA 93407, U.S.A.
(2) School of Mathematics, and Hunan Provincial Key Laboratory of Intelligent information processing and Applied Mathematics, Hunan University, Changsha, 410082, P.R. China

Summary: The de Branges-Rovnyak space $H(b)$ is generated by a bounded analytic function $b$ in the unit ball of $H^\infty$. When $b$ is a nonextreme point, the space $H(b)$ is invariant by the forward shift operator $M_z$. We show that the $H(b)$ spaces provide model spaces for expansive quasianalytic $2n$-isometric operators $T$ with $T^*T - I$ being rank one. Then we describe the invariant subspaces of the $2n$-isometric forward shift operator $M_z$ on $H(b)$.

Keywords: de Branges-Rovnyak space, $m$-isometry, operator model, shift invariant subspace

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