$C_0$-Semigroups on d.m.o. linear relations

This article develops a theory of one-parameter semigroups for linear relations, introducing the notion of domain-multivalued orthogonal (d.m.o.) relations.

We establish complete characterizations for generators of uniformly and strongly continuous semigroups, showing them to be maximal and densely d.m.o. relations, respectively.

Furthermore, the Hille-Yosida and Lumer-Phillips theorems are extended to this general setting, providing necessary and sufficient conditions for generating $C_0$-semigroups of contractions (identified as maximal anti-accumulative relations) and unitary operators (anti-selfadjoint relations).

This rigorous framework sheds new light on the peculiarities of one-parameter semigroups for linear relations.