Orbits under a class of isometries of $L^{1}[0,1]$

Terje Hõim¹

¹ Department of Mathematics Kent State University Kent, OH 44240, U.S.A.

Studia Mathematica, Tome 160 (2004) no. 1, pp. 1-22

Résumé

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.

Détail

DOI : 10.4064/sm160-1-1

Keywords: study orbits isometries certain class isometries set functions which orbit under isometry equivalent usual canonical basis mathinner ldotp ldotp ldotp dense set proof develop method get copies inside using geometric progressions method does disjoint relatively disjoint supports which seems common get copies method prove similar result shift operator leq infty finally study orbits multiplication operators mathbb set continuous complex valued functions mathbb absolutely convergent fourier series