On operators which attain their norm on every reducing subspace

In this talk, first, we discuss the spectral properties of operators that attain their norm on every reducing subspace. Next, we explore the structure of normal and quasinormal operators in this class. At the end, we describe paranormal operators whose adjoint is also paranormal. This gives direct proof of the fact that an operator is normal if and only if the operator and its adjoint are paranormal.