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

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.