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Proceedings of the American Mathematical Society

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

 

 

Reducing subspaces of contractions with no isometric part


Author: James Guyker
Journal: Proc. Amer. Math. Soc. 45 (1974), 411-413
MSC: Primary 47A15
DOI: https://doi.org/10.1090/S0002-9939-1974-0512617-9
MathSciNet review: 0512617
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Abstract: Let $ T$ be a contraction on a Hilbert space $ H$ and suppose that there is no nonzero vector $ f$ in $ H$ such that $ \vert\vert{T^n}f\vert\vert = \vert\vert f\vert\vert$ for every $ n = 1,2, \cdots $. In this paper, the reducing subspaces of $ T$ are characterized in terms of the range of $ 1 - {T^ \ast }T$. As a corollary, it is shown that $ T$ is irreducible if $ 1 - {T^ \ast }T$ has $ 1$-dimensional range. In particular, if $ U$ is the simple unilateral shift, then the restriction of $ {U^ \ast }$ to any invariant subspace for $ {U^ \ast }$ is irreducible.


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DOI: https://doi.org/10.1090/S0002-9939-1974-0512617-9
Keywords: Contraction with no isometric part, reducing subspace, irreducible contraction, unilateral shift
Article copyright: © Copyright 1974 American Mathematical Society