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BIRS Workshop Lecture Videos

Quasidense multifunctions Simons, Stephen


Quasidensity is a concept that can be applied to subsets of $E \times E^*$, where $E$ is a nonzero real Banach space. Every closed quasidense monotone set is maximally monotone, but there exist maximally monotone sets that are not quasidense. The graph of the subdifferential of a proper, convex lower semicontinuous function on $E$ is quasidense. The graphs of certain subdifferentials of certain nonconvex functions are also quasidense. (This follows from joint work with Xianfu Wang.) The closed monotone quasidense sets have a number of very desirable properties, including a sum theorem and a parallel sum theorem, and so quasidensity satisfies the ideal calculus rules. We give five conditions equivalent to the statement that a closed monotone set be quasidense, but quasidensity seems to be the only concept of the six that extends easily to nonmonotone sets. There are also generalizations to general Banach spaces of the Brezis-Browder theorem on linear relations, but we will not discuss these in this talk.

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