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Banach space operators with a bounded $$H^ \infty$$ functional calculus. (English) Zbl 0853.47010
Summary: We give a general definition for $$f(T)$$ when $$T$$ is a linear operator acting in a Banach space, whose spectrum lies within some sector, and which satisfies certain resolvent bounds, and when $$f$$ is holomorphic on a larger sector.
We also examine how certain properties of this functional calculus, such as the existence of a bounded $$H^\infty$$ functional calculus, bounds on the imaginary powers, and square function estimates are related. In particular we show that, if $$T$$ is acting in a reflexive $$L^p$$ space, then $$T$$ has a bounded $$H^\infty$$ functional calculus if and only if both $$T$$ and its dual satisfy square function estimates. Examples are given to show that some of the theorems that hold for operators in a Hilbert space do not extend to the general Banach space setting.

MSC:
 47A60 Functional calculus for linear operators