A multiplicative property characterizes quasinormal composition operators in -spaces

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