Cyclic multiplicity of a direct sum of forward and backward shifts

Gu, Caixing

doi:10.1090/proc/16301

Cyclic multiplicity of a direct sum of forward and backward shifts
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by Caixing Gu;

Proc. Amer. Math. Soc. 151 (2023), 2587-2601

DOI: https://doi.org/10.1090/proc/16301

Published electronically: March 21, 2023

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Abstract:

Let $E$ and $F$ be two complex Hilbert spaces. Let $S_{E}$ and $S_{F}$ be the shift operators on vector-valued Hardy space $H_{E}^{2}$ and $H_{F}^{2}$, respectively. We show that the cyclic multiplicity of $S_{E}\oplus S_{F} ^{\ast }$ equals $1+\dim E$. This result is classical when $\dim E=\dim F=1$ (see J. A. Deddens [On $A \oplus A^\ast$, 1972]; Paul Richard Halmos [A Hilbert space problem book, Springer-Verlag, New York-Berlin, 1982]; Domingo A. Herrero and Warren R. Wogen [Rocky Mountain J. Math. 20 (1990), pp. 445–466]). Our approach is inspired by the elegant and short proof of this classical result attributed to Nikolskii, Peller and Vasunin in Halmos’s book [A Hilbert space problem book, Springer-Verlag, New York-Berlin, 1982]. By using the invariant subspace theorems for $S_{E}\oplus S_{F}^{\ast }$ (see M. C. Câmara and W. T. Ross [Canad. Math. Bull. 64 (2021), pp. 98–111]; Caixing Gu and Shuaibing Luo [J. Funct. Anal. 282 (2022), 31 pp.]; Dan Timotin [Concr. Oper. 7 (2020), pp. 116–123]), we characterize non-cyclic subspaces of $S_{E}\oplus S_{F}^{\ast }$ when $\dim E<\infty$ and $\dim F=1$.

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