Reducing subspaces of contractions with no isometric part
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- by James Guyker PDF
- Proc. Amer. Math. Soc. 45 (1974), 411-413 Request permission
Abstract:
Let $T$ be a contraction on a Hilbert space $H$ and suppose that there is no nonzero vector $f$ in $H$ such that $||{T^n}f|| = ||f||$ 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.References
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Additional Information
- © Copyright 1974 American Mathematical Society
- 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