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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

On invariant linear manifolds


Author: P. A. Fillmore
Journal: Proc. Amer. Math. Soc. 41 (1973), 501-505
MSC: Primary 47A15
MathSciNet review: 0338804
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Abstract: For a linear transformation $ A$ on a Banach space, let $ \mathcal{L}(A)$ be the lattice of (not necessarily closed) invariant subspaces of $ A$. For $ A$ bounded it is shown that if $ \mathcal{L}(A \oplus A) \subset \mathcal{L}(T \oplus T)$, or if $ \mathcal{L}(A) \subset \mathcal{L}(T)$ and $ T$ commutes with $ A$, then $ T$ is a polynomial in $ A$. In the case of a Hilbert space, if $ \mathcal{L}(A) \subset \mathcal{L}({A^ \ast })$ then $ {A^ \ast }$ is a polynomial in $ A$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1973-0338804-X
PII: S 0002-9939(1973)0338804-X
Keywords: Invariant subspace lattice, locally algebraic operator
Article copyright: © Copyright 1973 American Mathematical Society