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Abstract
We consider conditions under which a continuous functional calculus for a Banach
space operator T Є L(X) may be extended to a bounded Borel functional calculus,
and under which a functional calculus for absolutely continuous (AC) functions
may be extended to one of for functions of bounded variation (BV). The natural
setting for investigating the former case is finitely spectral operators, and for the
latter, well-bounded operators.
Some such conditions are well-established. If X is a reflexive space, both type of
Extensions are assured; in fact if X contains an isomorphic copy of co, then every
Operator T Є L(X) that has a continuous functional calculus necessarily admits
a Borel one. We show that if a space X has a predual, then also every operator
T Є L(X) with a continuous functional calculus admits a bounded Borel functional
Calculus.
In case a Banach space X either contains an isomorphic copy of co, or has a
Predual, and T Є L(X) is an operator with an AC functional calculus, we find
that the existence of a decomposition of the identity of bounded variation for T
is sufficient to ensure that the AC functional calculus may be extended to a BV
functional calculus.
We also consider operators defined by a linear map on interpolation families of
Banach spaces [Xr, X∞] (r≥1), where for example Xp = lp, Lp[0,1] or Cp. We
show that under certain uniform boundedness conditions, the possession of a BV
functional calculus by operators on the spaces Xp, p Є (r, ∞), may be extrapolated
to the corresponding operators on the spaces Xr and X∞.