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The $H^{\infty}-$calculus and sums of closed operators

N. J. Kalton·L. Weis
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Abstract

We develop a very general operator-valued functional calculus for operators with an $H^{\infty}-$calculus. We then apply this to the joint functional calculus of two commuting sectorial operators when one has an $H^{\infty}$calculus. Using this we prove theorem of Dore-Venni type on sums of commuting sectorial operators and apply our results to the problem of $L_p-$maximal regularity. Our main assumption is the R-boundedness of certain sets of operators, and therefore methods from the geometry of Banach spaces are essential here. In the final section we exploit the special Banach space structure of $L_1-$spaces and $C(K)-$spaces, to obtain some more detailed results in this setting.

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N. J. Kalton, L. Weis. 2000-10-15. The $H^{\infty}-$calculus and sums of closed operators. https://arxiv.org/abs/math/0010155

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