Adjoints of semigroups of linear operators in Banach spaces (Q1323114)

The authors prove the complex generalization of the well-known Phillips theorem on the adjoint semigroups for a recently studied new class of semigroups [\textit{I. Miyadera} and \textit{N. Tanaka}, J. Math. Anal. Appl. 143, No. 2, 358-378 (1989; Zbl 0697.47039)], precisely semigroups \(\{T(t): T\geq 0\}\) on a Banach space \(X\) satisfying two conditions: (a) \(T(t) C= CT(t)\) for \(t>0\); (b) \(\text{Range} (C)\subset \Sigma:= \{x\in X: \lim_{t\to 0+} T(t) x=x\}\), where \(C\) is an injective bounded linear operator in \(X\) with dense range. As applications the authors give the description of the adjoint semigroups of both semigroups of growth order \(\alpha\) and of semigroups of class \((C_{(k)})\) already in a unified way.