For any linear operator T acting on a Hilbert space H, its Aluthge transform T˜ where T˜ = |T| 1 2U|T| 1 2 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 dierent 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.

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