Quasinormal extensions of subnormal operator-weighted composition operators in ℓ2-spaces

We prove the subnormality of an operator-weighted composition operator whose symbol is a transformation of a discrete measure space and whose weights are multiplication operators in L2-spaces, under the assumption of existence of a family of probability measures whose Radon-Nikodym derivatives behave regular along the trajectories of the symbol. We build the quasinormal extension, which is a weighted composition operator induced by the same symbol. We give auxiliary results concerning commutativity of operator-weighted composition operators with multiplication operators.