On the asymptotic property of normal operators commutators in weak, strong and the strongest $B$-module operator typologies

The asymptotic variant of Fugled–Putnam classic theorem is proved for the pair of limited normal operator acting in the Hilbert $B$-module $E$ in weak, strong and the strongest $B$-module operator topologies. Earlier, the asymptotic variants of the above mentioned theorem have been obtained in the papers [1–4].