For any linear operator T acting on a Hilbert space H, its Aluthge transform T˜ where
T˜ = jTj
is another linear operator on H: It is known that T˜ preserves the spectral
properties of T and more importantly that T has a non trivial closed invariant subspace
if and only if T has. In this project Aluthge transforms of di erent classes of operators
in Hilbert spaces were studied. In addition, generalized Aluthge transforms, as well as
powers of Aluthge transformations were sort and looked at. Lastly, the numerical range of
T was discussed but for some classes of operators.