Optimal domain and integral extension of operators

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Operator theory and functional analysis have a long tradition, initially being guided by problems from mathematical physics and applied mathematics. Much of the work in Banach spaces from the 1930s onwards resulted from investigating how much real (and complex) variable function theory might be extended to fu- tions taking values in (function) spaces or operators acting in them. Many of the ? rst ideas in geometry, basis theory and the isomorphic theory of Banach spaces have vector measure-theoretic origins and can be credited (amongst others) to N. Dunford, I. M. Gelfand, B. J. Pettis and R. S. Phillips. Somewhat later came the penetratingcontributionsofA. Grothendieck, whichhavepervadedandin? uenced theshapeoffunctionalanalysisandthetheoryofvectormeasures/integrationever since. Today, each of the areas of functional analysis/operator theory, Banach spaces, and vector measures/integration is a strong discipline in its own right. However, it is not always made clear that these areas grew up together as cousins and that they had, and still have, enormous in? uences on one another. One of the aims of this monograph is to reinforce and make transparent precisely this important point.