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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


Irreducible semigroups of functionally positive nilpotent operators

Author: Yong Zhong
Journal: Trans. Amer. Math. Soc. 347 (1995), 3093-3100
MSC: Primary 47A15; Secondary 47B38, 47D03
MathSciNet review: 1264835
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Abstract: For each irrational number $ \theta \in (0,1)$, we construct a semigroup $ {\mathcal{S}_\theta }$ of nilpotent operators on $ {\mathcal{S}^2}([0,1])$ that are also partial isometries and positive in the sense that the operator maps nonnegative functions to nonnegative functions. We prove that each semigroup $ {\mathcal{S}_\theta }$ is discrete in the norm topology and hence norm-closed and that the weak closure of $ {\mathcal{S}_\theta }$ is independent of $ {\mathcal{S}_\theta }$. We show that each semigroup $ {\mathcal{S}_\theta }$ has no nontrivial invariant subspaces.

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PII: S 0002-9947(1995)1264835-0
Keywords: Invariant subspace, nilpotent operator, semigroup of operators
Article copyright: © Copyright 1995 American Mathematical Society