Ian Doust's information pages

Extensions of an AC(σ) functional calculus

J. Math. Anal. Appl. 362 (2010), 100-106. [doi:10.1016/j.jmaa.2009.10.001.]

Ian Doust and Venta Terauds

Address: (ID) School of Mathematics, University of New South Wales, Sydney NSW 2052, Australia

Email:  i.doust@unsw.edu.au

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.

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