DR. IMAGIRI STANLEY KATHURIMA

Every reducible operator can be decomposed into normal and completely non-normal operators.
Unfortunately, there are several non normal operators which are irreducible. However, every
operator whose self-commutator is bounded, is reducible. Putnam’s inequality implies boundedness
of the self-commutator for hyponormal operators. In this paper, the Putnam’s inequality is
studied for n-Power normal, n-power quasinormal and w-hyponormal operators.

Every reducible operator can be decomposed into normal and completely non-normal operators.
Unfortunately, there are several non normal operators which are irreducible. However, every operator
whose self-commutator is bounded, is reducible. Berger-Shaw inequality implies boundedness
of the trace of the self-commutator for hyponormal operators. In this paper, the Berger-Shaw
inequality is studied for n-Power normal, n-power quasinormal and w-hyponormal operators.

In this paper, infinite-Power normal and infinite-Power quasinormal operators are introduced. Amongst
other results, it is proved that, infinite-Power normal operators have translation invariant property
and that, if any operator and its adjoint happens to be n-Power quasinormal, then such an
operator becomes normal

In this monograph, via Aluthge transformations, it is proved that every n-Power normal operator
is normaloid and through paranormality, w-hyponormal operators are shown to be indepedent
from n-Power normal, hence from n-Power quasinormal operators. Nevertheless, an attempt to
throw more materials- rather than the class of normal operators-, in the intersection of these
non-normal classes is presented.

Reducibility implies direct sum decompositions of Hilbert space operators and any pair of operators
which satisfy the Putnam-Fuglede theorem is reducible. In this presentation, the familiar
Putnam-Fuglede theorem is firstly investigated for n-Power normal operators. Then, it’s assymetric
version is studied for n-Power normal and w-hyponormal operators. As a consequence,
more conditions implying normality, or even similarlity between these two operator classes, are
deduced via this theorem.