Wednesday, 4 February 2009

books - reference for weak*-semigroup

I don't quite follow your notation, but I'll answer what I think you might be asking.



A C0 or strongly continuous semigroup of operators Tt on a Banach space X is one such that Ttxtox in norm as tto0, i.e. ||Ttxx||Xto0. In other words, TttoI in the strong operator topology.



A weakly continuous semigroup Tt has Ttxtox weakly as tto0, i.e. f(Ttx)tof(x) for each finX. In other words, TttoI in the weak operator topology.



In fact, these two conditions are equivalent. This appears as Theorem 1.6 of K.-J. Engel and R. Nagel, A Short Course on Operator Semigroups.



So this is why you never hear anyone talking about weakly continuous semigroups.

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