Invariant subspaces of the generalized backward shift operator and rational functions

Ivanova, O.; Melikhov, S.; Melikhov, Yu.

doi:10.1090/spmj/1734

Invariant subspaces of the generalized backward shift operator and rational functions
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by O. A. Ivanova, S. N. Melikhov and Yu. N. Melikhov
Translated by: A. Tselishchev

St. Petersburg Math. J. 33 (2022), 927-942

DOI: https://doi.org/10.1090/spmj/1734

Published electronically: October 31, 2022

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Abstract:

The paper is devoted to a complete characterization of proper closed invariant subspaces of the generalized backward shift operator (Pommiez operator) in the Fréchet space of all holomorphic functions in a simply connected domain $\Omega \ni 0$ in the complex plane. In the case when the function that generates this operator does not have zeros in $\Omega$, all such subspaces are finite-dimensional. If in addition $\Omega$ coincides with the entire complex plane, then the generalized backward shift operator under consideration is unicellular. If this function has zeros in $\Omega$, then the above family of invariant subspaces splits into two classes: the first consists of finite-dimensional subspaces, and the second of infinite-dimensional ones.

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