A dynamical system $f:X\to X$ is said to be topologically transitive if for any two nonempty open sets $U,V$ there exists $n \in \mathbb{Z}$ such that $f^{\circ n}(U) \cap V \neq \emptyset$. The following two definitions of topological weak mixing can be found in the literature:

- $f$ is topologically weak mixing if $f\times f$ is topologically transitive on $X\times X$ (e.g here)
- $f$ is topologically weak mixing if it has no non-constant continuous (with respect to the topology) eigenfunctions of the shift operator. (e.g. here)

How are these definitions related? Can they be shown to be equivalent?