We study the orbits of isometries of L¹[0,1]. For a certain class of isometries we show that the set of functions f in L¹[0,1] for which the orbit of f under the isometry T is equivalent to the usual canonical basis e₁,e₂,e₃,... of l¹ is an open dense set. In the proof we develop a new method to get copies of l¹ inside L¹[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 , 1 ≤ p < ∞. Finally, we study the orbits of multiplication operators on H² and A(), the set of all continuous complex-valued functions on with absolutely convergent Fourier series.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm160-1-1,
author = {Terje H\~oim},
title = {Orbits under a class of isometries of L$^1$[0,1]},
journal = {Studia Mathematica},
volume = {162},
year = {2004},
pages = {1-22},
zbl = {1067.47042},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm160-1-1}
}
Terje Hõim. Orbits under a class of isometries of L¹[0,1]. Studia Mathematica, Tome 162 (2004) pp. 1-22. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm160-1-1/