2 problems

Let $(X,\mathscr{B},\mu,T)$ be a measure-preserving system, let $U$ denote its Koopman operator, and let $f\in L^2(X,\mu)$ satisfy … for every $g\in L^2(X,\mu)$. A sequence…

Let $(X,\mu,T)$ be a measure-preserving system, and let $\Phi_N=\{1,\dots,N\}$ be the standard Følner sequence in $\mathbb{Z}$. For a fixed positive integer $k$, say that $X$ has p…