The infinite product of contraction semigroups on l^{1}({mathbb {N}}) and l^{infty }({mathbb {N}})

Abstract

I provide an example of a family of commuting contraction semigroups (e^{t B_{n}})_{nin {mathbb {N}}} defined on l^{1}({mathbb {N}}) such that the product semigroup prod _{n=1}^{infty }e^{tB_{n}} exists and has bounded generator. The infinite product of the corresponding family of adjoint semigroups (e^{t B_{n}^{*}})_{nin {mathbb {N}}} defined on l^{infty }({mathbb {N}}) also exists and its generator is bounded. I give explicit formulae for these generators. The results follow from a general convergence theorem for such semigroups proved in Arendt et al. (J Funct Anal 160: 524–542, 1998).