Graduate Texts in Mathematics bridge the gap between passive study and creative understanding, offering graduate-level introductions to advanced topics in mathematics. The volumes are carefully written as teaching aids and highlight characteristic features of the theory. Although these books are frequently used as textbooks in graduate courses, they are also suitable for individual study.More information about this series at

Abstract. We develop a very general operator-valued functional calculus for operators with an H ∞ −calculus. We then apply this to the joint functional calculus of two commuting sectorial operators when one has an H ∞ 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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Abstract. We show that when a linear quotient map to a separable Banach space X has a Lipschitz right inverse, then it has a linear right inverse. If a separable space X embeds isometrically into a Banach space Y , then Y contains an isometric linear copy of X. This is false for every nonseparable weakly compactly generated Banach space X. Canonical examples of nonseparable Banach spaces which are Lipschitz isomorphic but not linearly isomorphic are constructed. If a Banach space X has the bounded approximation property and Y is Lipschitz isomorphic to X, then Y has the bounded approximation property.

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