The quaternionic evolution operator (Q549210)

The authors study quaternionic semigroups and groups generated by quaternionic, in general unbounded, linear operators of the form \(T=T_0+i T_1+j T_2+k T_3\). The components \(T_l\), \(l=0,1,2,3\), do not necessarily commute. The quaternionic version of the Hille-Phillips-Yosida generation theorem is established. This result is based on the fact that the Laplace transform of the quaternionic semigroup \(e^{tT}\) is the \(S\)-resolvent operator \((T^2-2 \text{{Re}[s]}T+| s|^2 J)^{-1} ({\overline s}J-T)\), the quaternionic analogue of the classical resolvent operator. Here, \(J\) denotes the quaternionic identity operator.