Dilations of commuting C0-semigroups with bounded generators and the von Neumann polynomial inequality

Consider d commuting C0-semigroups (or equivalently: d-parameter C0-semigroups) over a Hilbert space for d∈N. In the literature (cf. [31], [27], [28], [24], [18], [26]), conditions are provided to classify existence of unitary and regular dilations. Some these require inspecting values semigroups, some provide only sufficient conditions, others involve verifying sophisticated properties generators. By focussing on semigroups with bounded generators, we establish simple natural condition viz. complete dissipativity, which naturally extends basic notion dissipativity Using examples non-doubly this property can be shown strictly stronger than dissipativity. As first main result, demonstrate that completely characterises dilations, extend case arbitrarily many C0-semigroups. We furthermore show all multi-parameter (with generators) admit weaker norm criteria The paper concludes an application von Neumann polynomial inequality problem, formulate semigroup setting solve negatively d⩾2.