This is an announcement for the paper "Extensions of an $AC(\sigma)$ functional calculus" by Ian Doust and Venta Terauds.
Abstract: On a reflexive Banach space $X$, if an operator $T$ admits a functional calculus for the absolutely continuous functions on its spectrum $\sigma(T) \subseteq \mathbb{R}$, then this functional calculus can always be extended to include all the functions of bounded variation. This need no longer be true on nonreflexive spaces. In this paper, it is shown that on most classical separable nonreflexive spaces, one can construct an example where such an extension is impossible. Sufficient conditions are also given which ensure that an extension of an $\AC$ functional calculus is possible for operators acting on families of interpolation spaces such as the $L^p$ spaces.
Archive classification: math.FA
Mathematics Subject Classification: 47B40
The source file(s), extns-f-submit.tex: 36353 bytes, is(are) stored in gzipped form as 0803.2131.gz with size 11kb. The corresponding postcript file has gzipped size 84kb.
Submitted from: i.doust@unsw.edu.au
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/0803.2131
or
http://arXiv.org/abs/0803.2131
or by email in unzipped form by transmitting an empty message with subject line
uget 0803.2131
or in gzipped form by using subject line
get 0803.2131
to: math@arXiv.org.