A multiplicative property characterizes quasinormal composition   operators in $L^2$-spaces
  A densely defined composition operator in an $L^2$-space induced by a measurable transformation $\phi$ is shown to be quasinormal if and only if the Radon-Nikodym derivatives $h_{\phi^n}$ attached to powers $\phi^n$ of $\phi$ have the multiplicative property: $h_{\phi^n} = h_{\phi}^n$ almost everywhere for n = 0, 1, 2, .... 
Piotr Budzynski, Zenon Jan Jablonski, Il Bong Jung, Jan Stochel
math.FA