Joint subnormality of $n$-tuples and $C_0$-semigroups of composition operators on $L^2$-spaces, II
In the previous paper, we have characterized (joint) subnormality of a $C_0$-semigroup of composition operators on $L^2$-space by positive definiteness of the Radon–Nikodym derivatives attached to it at each rational point. In the present paper, we show that in the case of $C_0$-groups of composition operators on $L^2$-space the positive definiteness requirement can be replaced by a kind of consistency condition which seems to be simpler to work with. It turns out that the consistency condition also characterizes subnormality of $C_0$-semigroups of composition operators on $L^2$-space induced by injective and bimeasurable transformations. The consistency condition, when formulated in the language of the Laplace transform, takes a multiplicative form. The paper concludes with some examples.