Alternative definition of (strongly?, weakly?) mixing

Question

In his paper On Automorphisms of Compact Groups, Halmos defines a measure preserving transformation $\alpha$ to be ergodic if the only complex-valued measure preserving solutions of the equation $f^\alpha = f$ are constant almost everywhere and to be mixing if the only measure preserving solutions of the $f^\alpha = \lambda f$ for any constant $\lambda \in \Bbb{C}$ are constant almost every where (implying that actually $\lambda = 1$). (Here $f^\alpha(x) = f(\alpha(x))$.) He refers to Hopf's Ergodentheorie for proofs that these definitions are equivalent to the more common ones. I am familiar with the proof that ergodic in this sense is equivalent to ergodicity as defined by a $0$-$1$ law for $\alpha$-invariant sets. I don't have access to Hopf's book and have two questions: (1) is the given definition of mixing equivalent to strongly mixing or weakly mixing? (2) can anyone point me at a more accessible reference to the proof that this definition of mixing is equivalent to the one of the more common definitions (e.g., as given in Halmos's Lectures on Ergodic Theory), or, even, if it's not too long, post a proof here.