Relatively tauberian resolvent operators (Q1924513)

Let \(X\) be a complex Banach space. We say that a closed linear operator \(A\) generates a uniformly bounded resolvent family (UBR), if \(\{\alpha\in R: \alpha >0\} \subset \rho (A)\) and there is a constant \(K>0\) such that \(|\alpha (\alpha I-A)^{-1} |\leq K\) for every \(\alpha>0\). Let \(T\) be a bounded operator on \(X\) which is injective and let \(V\) be a total subspace of \(X^*\). Definition. The operator \(T\) is \(V\)-relatively tauberian, if \(T^{**}F\in\widehat X\) implies that \(F\in \widehat X+ V^\perp\). If \(V=(T^* (X^*))^-\), then \(T\) is relatively tauberian. If \(V=X^*\), \(T\) is called tauberian. Let \(A_\alpha\) be the operator \(\alpha A(\alpha I-A)^{-1}\) for every \(\alpha>0\). We denote \(D'(A)= \{x\in X: \{A_\alpha x\}_{\alpha >0}\) is bounded\}. The author proves the following theorem: Assume that \(A\) generates a UBR. Then \(D'(A)= D(A)\) if and only if \((\alpha I-A)^{-1}\) is relatively tauberian for some \(\alpha>0\).