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

By Nigel J. Kalton, Lutz Weis. 2000

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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 sectorial operators when one has an $H^{\infty}$calculus. Using this we prove theorem of Dore-Venni type on sums of 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.

Subfields: Applied Mathematics, Mathematical Physics

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