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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Adjoint abelian operators on $ L\sp{p}$ and $ C(K)$


Authors: Richard J. Fleming and James E. Jamison
Journal: Trans. Amer. Math. Soc. 217 (1976), 87-98
MSC: Primary 47B37
DOI: https://doi.org/10.1090/S0002-9947-1976-0394279-3
MathSciNet review: 0394279
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Abstract: An operator A on a Banach space X is said to be adjoint abelian if there is a semi-inner product $ [ \cdot , \cdot ]$ consistent with the norm on X such that $ [Ax,y] = [x,Ay]$ for all $ x,y \in X$. In this paper we show that every adjoint abelian operator on $ C(K)$ or $ {L^p}(\Omega ,\Sigma ,\mu )\;(1 < p < \infty ,p \ne 2)$ is a multiple of an isometry whose square is the identity and hence is of the form $ Ax( \cdot ) = \lambda \alpha ( \cdot )(x \circ \phi )( \cdot )$ where $ \alpha $ is a scalar valued function with $ \alpha ( \cdot )\alpha \circ \phi ( \cdot ) = 1$ and $ \phi $ is a homeomorphism of K (or a set isomorphism in case of $ {L^p}(\Omega ,\Sigma ,\mu ))$ with $ \phi \circ \phi =$ identity (essentially).


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DOI: https://doi.org/10.1090/S0002-9947-1976-0394279-3
Keywords: Adjoint abelian, isometries, semi-inner products, reflections, scalar operators
Article copyright: © Copyright 1976 American Mathematical Society