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We study the notion of dual quasisemigroups of bounded linear operators as a generalization of that for strongly continuous semigroup and prove some properties similar to the dual of a semigroup, among other things we prove that for reflexive Banach spaces the dual quasisemigroup is strongly continuous on (0, +∞). This allows us to extend some recent criteria of controllability to a general class of evolution equations in reflexive Banach spaces.

Using some techniques from vector integration, we prove the weak measurability of the adjoint of strongly continuous semigroups which factor through Banach spaces without isomorphic copy ofl1; we also prove the strong continuity away from zero of the adjoint if the semigroup factors through Grothendieck spaces. These results are used, in particular, to characterize the space of strong continuity of{T**(t)}t≥0, which, in addition, is also characterized for abstractL- andM-spaces. As a corollary, it is proven that abstractL-spaces with no copy ofl1are finite-dimensional.

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