A characterization of sun-reflexivity (Q910690)

The dual semigroup \(\{T^*(t)\}_{t\geq 0}\) of a strongly continuous semigroup \(\{T(t)\}_{t\geq 0}\) of bounded linear operators (i.e. \(C_ 0\)-semigroup) on a Banach space X need not be strongly continuous in \(X^*\). Let \(X^{\odot}\) denote the maximal invariant subspace for \(T^*(\cdot)\) such that the restriction \(T^{\odot}(\cdot)=T^*(\cdot)|_{X^{\odot}}\) is a \(C_ 0\)- semigroup. Let j be the canonical embedding of X into \(X^{**}\), and \(\gamma_{\odot}\) be the restriction mapping, sending each \(x^{**}\in X^{**}\) to \(x^{**}| X^{\odot}\). Then the mapping \(j_{\odot}=\gamma_{\odot}\circ j\) is a norm isomorphism form X into \(X^{\odot \odot}\). In general, it is not onto. If it is, X is called reflexive. It is known [cf. \textit{E. Hille} and \textit{R. S. Phillips}, Functional Analysis and Semigroups, Am. Math. Soc. Colloquium Pubs. vol. 31 (1957; Zbl 0078.100)] that \(\odot\)-reflexivity of X is equivalent to \(\sigma (X,X^{\odot})\)-compactness of \((\lambda -A)^{-1}\), \(\lambda\in \rho (A)\), where A denotes the infinitesimal generator of T(\(\cdot)\). The present paper shows that it is also equivalent to the weak compactness of \((\lambda -A)^{-1}\), and, whenever X has the Dunford-Pettis property (e.g. \(L^ 1\)-spaces), \(\odot\)-reflexivity of X is even equivalent to the compactness of \((\lambda -A)^{-1}\).