Orbits under a class of isometries of $L^{1}[0,1]$
Volume 160 / 2004
Abstract
We study the orbits of isometries of $L^{1}[0,1]$. For a certain class of isometries we show that the set of functions $f$ in $L^{1}[0,1]$ for which the orbit of $f$ under the isometry $T$ is equivalent to the usual canonical basis $\{ e_{1}, e_{2}, e_{3}, \mathinner {\ldotp \ldotp \ldotp }\} $ of $l^{1}$ is an open dense set. In the proof we develop a new method to get copies of $l^{1}$ inside $L^{1}[0,1]$ using geometric progressions. This method does not use disjoint or relatively disjoint supports, which seems to be the most common way to get such copies. We also use this method to prove a similar result for the shift operator on $l^{p}$, $1 \leq p < \infty $. Finally, we study the orbits of multiplication operators on $H^{2}$ and $A({{\mathbb T}})$, the set of all continuous complex-valued functions on ${{\mathbb T}}$ with absolutely convergent Fourier series.
