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Transactions of the American Mathematical Society

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

 

 

On positive contractions in $ L\sp{p}$-spaces


Author: H. H. Schaefer
Journal: Trans. Amer. Math. Soc. 257 (1980), 261-268
MSC: Primary 47B55
DOI: https://doi.org/10.1090/S0002-9947-1980-0549167-7
MathSciNet review: 549167
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Abstract: Let T denote a positive contraction $ (T\, \geqslant \,0,\,\left\Vert T \right\Vert\, \leqslant \,1)$ on a space $ {L^p}(\mu )\,(1\, < \,p\, < \, + \,\infty )$. A primitive nth root of unity $ \varepsilon $ is in the point spectrum $ P\sigma (T)$ iff it is in $ P\sigma (T')$; if so, the unimodular group generated by $ \varepsilon $ is in both $ P\sigma (T)$ and $ P\sigma (T')$. In turn, this is equivalent to the existence of n-dimensional Riesz subspaces of $ {L^p}$ and $ {L^q}({p^{ - \,1}}\, + \,{q^{ - \,1}}\, = \,1)$ which are in canonical duality and on which T (resp., $ T'$) acts as an isometry. If, in addition, T is quasi-compact then the spectral projection associated with the unimodular spectrum of T (resp., $ T'$) is a positive contraction onto a Riesz subspace of $ {L^p}$ (resp., $ {L^q}$) on which T (resp., $ T'$) acts as an isometry.


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DOI: https://doi.org/10.1090/S0002-9947-1980-0549167-7
Article copyright: © Copyright 1980 American Mathematical Society