Extensions of an $AC(σ)$ functional calculus

Details Extensions of an $AC(σ)$ functional calculus Extensions of an $AC(σ)$ functional calculus

2008-03-14

arxiv.org/abs/0803.2131v1

J. Math. Anal. Appl. 362 (2010), 100-106

Mathematics

Functional Analysis

Scientific paper

10.1016/j.jmaa.2009.10.001

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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