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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.

47A60 Functional calculus for linear operators