### Abstract:

Spectral theory of linear operators on Hilbert spaces is a pillar in several developments
in mathematics, physics and quantum mechanics. Its concepts like the spectrum of a linear
operator, eigenvalues and vectors, spectral radius, spectral integrals among others have
useful applications in quantum mechanics.
In this work we study the properties of a large subclass of bounded linear operators
on a Hilbert space H, which is posinormal operators. This class was first introduced
by Rhaly (see [13]). A bounded linear operator T on a Hilbert space H is said to be
posinormal if there exist a positive operator P, such that
T T∗ = T
∗P T.
Also, We have discussed the relation between posinormal operators and other classes of
bounded linear operators.
The first chapter contains an introduction to our work. We have presented some fundamental properties, for example: Hilbert spaces, bounded linear operators, spectrum of
bounded linear operator and we have also defined some classes of bounded linear operators.
In the second chapter, we presented several proprieties of posinormal operators. Also,
we showed the deferences between the deferent classes of bounded linear operators, and
we finished the chapter by studying the spectrum of posinormal operators.
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In the last chapter, we studied the powers of posinormal operators, more precisely, we
have shown the relation between a posinormal operator and its powers.