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First, some classic properties of a weighted Frobenius-Perron operator ${𝒫}_{\varphi }^u$ on $L^1\left(\Sigma \right)$ as a predual of weighted Koopman operator $W=u{U}_{\varphi }$ on $L^{\infty }\left(\Sigma \right)$ will be investigated using the language of the conditional expectation operator. Also, we determine the spectrum of ${𝒫}_{\varphi }^u$ under certain conditions.

Some stronger and equivalent metrics are defined on $ℳ$, the set of all bounded normal operators on a Hilbert space $H$ and then some topological properties of $ℳ$ are investigated.