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Research Papers
Do some nontrivial closed $z$-invariant subspaces have the division property?
J. Esterle IMB, UMR 5251, Université de Bordeaux 351, cours de la Libération, 33405 - Talence, France
Abstract:
Banach spaces $E$ of functions holomorphic on the open unit disk $\mathbb{D}$ are considered such that the unilateral shift $S$ and the backward shift $T$ are bounded on $E$. Under the assumption that the spectra of $S$ and $T$ are equal to the closed unit disk, the existence is discussed of closed $z$-invariant subspaces $N$ of $E$ having the “division property,” which means that the function $f_{\lambda}\colon z \mapsto {f(z)\over z-\lambda}$ belongs to $N$ for every $\lambda \in \mathbb{D}$ and for every $f \in N$ with $f(\lambda)=0$. This question is related to the existence of nontrivial bi-invariant subspaces of Banach spaces of hyperfunctions on the unit circle $\mathbb{T}$.
Keywords:
unilateral shift, backward shift, division property, invariant subspace, bi-invariant subspace.
Received: 05.05.2020
Citation:
J. Esterle, “Do some nontrivial closed $z$-invariant subspaces have the division property?”, Algebra i Analiz, 33:4 (2021), 173–209; St. Petersburg Math. J., 33:4 (2022), 711–738
Linking options:
https://www.mathnet.ru/eng/aa1775 https://www.mathnet.ru/eng/aa/v33/i4/p173
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Abstract page: | 113 | Full-text PDF : | 11 | References: | 33 | First page: | 10 |
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