On operators which attain their norm on every reducing subspace

In this article, first, we study the spectral properties of operators which attain their norm on every reducing subspace and then we study the structure of normal and quasinormal operators in this class. At the end we give a description of paranormal operators whose adjoint is also paranormal. This gives a direct proof of the fact that an operator is normal if and only if the operator and its adjoint are paranormal. © 2022, Tusi Mathematical Research Group (TMRG).