On positive contractions in $L^{p}$-spaces
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- by H. H. Schaefer PDF
- Trans. Amer. Math. Soc. 257 (1980), 261-268 Request permission
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 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.References
-
N. Dunford and J. T.Schwartz, Linear operators, vol. I, Wiley-Interscience, New York, 1958.
- Michael Lin, Quasi-compactness and uniform ergodicity of positive operators, Israel J. Math. 29 (1978), no. 2-3, 309–311. MR 493502, DOI 10.1007/BF02762018
- Helmut H. Schaefer, 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, Ergodic properties of nonnegative matrices. II, Pacific J. Math. 26 (1968), 601–620. MR 236745, DOI 10.2140/pjm.1968.26.601
Additional 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