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Proceedings of the American Mathematical Society

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Two types of hyperinvariant subspaces


Author: Robert M. Kauffman
Journal: Proc. Amer. Math. Soc. 39 (1973), 553-558
MSC: Primary 47A15; Secondary 47B40
MathSciNet review: 0336389
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Abstract: Let $ A$ be a bounded operator in a Banach space $ B$. Suppose that $ A$ has the single valued extension property. Given a closed set $ F$ in the complexes, define $ {\sigma _A}(F)$ to be the set of all $ x$ in $ B$ such that there is an analytic function $ x(\lambda )$ from the complement of $ F$ to $ B$ with $ (A - \lambda I)x(\lambda ) = x$. $ A$ is said to have property $ Q$ if $ {\sigma _A}(F)$ is a closed subset of $ B$ for every $ F$.

Let $ A$ be, again, a bounded operator in a Banach space $ B$. Given a real number $ b$, define $ {S_A}(b)$ to be the set of all $ x$ in $ B$ such that $ \exp ( - ct)\exp (At)x$ is a bounded function from the nonnegative reals to $ B$ for all $ c > b$. $ A$ is said to have property $ \operatorname{P} $ if $ {S_A}(b)$ is a closed subspace of $ B$ for all $ b$.

These two properties are discussed in this paper.


References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0336389-5
Keywords: Hyperinvariant subspace, single valued extension property, spectral operator, quasinilpotent operator, hyponormal operator
Article copyright: © Copyright 1973 American Mathematical Society