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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Classification of nearly invariant subspaces of the backward shift


Author: Eric Hayashi
Journal: Proc. Amer. Math. Soc. 110 (1990), 441-448
MSC: Primary 47A15; Secondary 30H05, 47B35, 47B38
DOI: https://doi.org/10.1090/S0002-9939-1990-1019277-0
MathSciNet review: 1019277
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Abstract: Let $ {S^*}$ denote the backward shift operator on the Hardy space $ {H^2}$ of the unit disk. A subspace $ M$ of $ {H^2}$ is called nearly invariant if $ {S^*}h$ is in $ M$ whenever $ h$ belongs to $ M$ and $ h(0) = 0$. In particular, the kernel of every Toeplitz operator is nearly invariant. A function theoretic characterization is given of those nearly invariant subspaces which are the kernels of Toeplitz operators, and it is shown that they can be put into one-to-one correspondence with the Cartesian product of the set of exposed points of the unit ball of $ {H^1}$ with the set of inner functions.


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DOI: https://doi.org/10.1090/S0002-9939-1990-1019277-0
Article copyright: © Copyright 1990 American Mathematical Society