Semigroups with nondensely defined generating operator
In the present paper, semigroups \(\{U(t), t\geq0\}\) of class \(A(\varphi)\), more general than those of class \(A(\alpha, \beta)\) [\textit{Yu. T. Sil'chenko}, Funct. Anal. Appl. 33, No. 4, 315--317 (1999), translation from Funkts. Anal. Prilozh. 33, No. 4, 90--93 (1999; Zbl 0949.34047); Nonlinear Anal., Theory Methods Appl. 36, No. 3(A), 345--352 (1999; Zbl 1002.34051)], are studied in a Banach space \(E\). Namely, the semigroup \(\{U(t), t\geq0\}\) is said to be of class \(A(\varphi)\) if for \(D\) a linear set in \(E\) and \(\varphi(t)\) a continuous function in \(t>0\), the following conditions are fulfilled: (1) \(U(t)\colon E\to D\); (2) \(U(t_1+t_2)=U(t_1)U(t_2)\) for \(t_1, t_2>0\); (3) \(\lim_{t\to0}U(t)x = x\) for all \(x\in D\); (4) \(\|U(t)\|\leq\varphi(t)\). Properties of these semigroups such as strong continuity, relations between \(D\), \(D(U'(0))\), \(D(\overline{U'(0)})\) and the existence of the resolvent of the generator are proved. Some examples are given.
scientific article; zbMATH DE number 5155810