On the adjoint of a strongly continuous semigroup.(English)Zbl 1165.47026

In this nicely written paper, the authors prove the weak measurability of the adjoint of strongly continuous semigroups which factor through Banach spaces without isomorphic copy of $$l_1$$. They obtain the strong continuity away from zero of the adjoint if the semigroup factors through Grothendieck spaces.
As consequences of the main results, the authors characterize the strong continuity of $$\{T^{**}(t)\}_{t\geq 0}$$, which is also described for abstract $$L$$- and $$M$$-spaces. As a corollary, they prove that abstract $$L$$-spaces with no copy of $$l_1$$ are finite-dimensional.

MSC:

 47D06 One-parameter semigroups and linear evolution equations
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References:

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