Quasinormal extensions of subnormal operator-weighted composition   operators in $\ell^2$-spaces
  We prove the subnormality of an operator-weighted composition operator whose symbol is a transformation of a discrete measure space and weights are multiplication operators in $L^2$-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. 
Piotr Budzynski, Piotr Dymek, Artur Planeta
math.FA