## On positive contractions in $L^{p}$-spaces

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- by H. H. Schaefer
- Trans. Amer. Math. Soc.
**257**(1980), 261-268 - DOI: https://doi.org/10.1090/S0002-9947-1980-0549167-7
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## Abstract:

Let*T*denote a positive contraction $(T \geqslant 0, \left \| T \right \| \leqslant 1)$ on a space ${L^p}(\mu ) (1 < p < + \infty )$. A primitive

*n*th 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.

## References

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*Banach lattices and positive operators*, Die Grundlehren der mathematischen Wissenschaften, Band 215, Springer-Verlag, New York-Heidelberg, 1974. MR**0423039**, DOI 10.1007/978-3-642-65970-6 - D. Vere-Jones,
*Ergodic properties of nonnegative matrices. I*, Pacific J. Math.**22**(1967), 361–386. MR**214145**, DOI 10.2140/pjm.1967.22.361 - D. Vere-Jones,
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*Linear operators*, vol. I, Wiley-Interscience, New York, 1958.

## Bibliographic Information

- © Copyright 1980 American Mathematical Society
- 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