A multiplicative property characterizes quasinormal composition operators in L2-spaces

A densely defined composition operator in an L2-space induced by a measurable transformation ϕ is shown to be quasinormal if and only if the Radon-Nikodym derivatives hϕn attached to powers ϕn of ϕ have the multiplicative property:  hϕn=hϕn almost everywhere for n=0,1,2,….