Compact decoupling for an abstract system of thermoelasticity of type III

In this note, we show that the difference between the C0-semigroups generated by an abstract system of thermoelasticity of type III and a suitable decoupled system is compact in the underlying natural energy space. The problem is reduced to prove the norm continuity of this difference and the compactness of the difference between the resolvent of their generators. The argument is based on an operator semigroup technique.