In this paper, we establish some upper bounds of the numerical radius of a bounded linear operator defined on a complex Hilbert space with polar decomposition , involving generalized Aluthge transform. These bounds generalize some bounds of the numerical radius existing in the literature. Moreover, we consider particular cases of generalized Aluthge transform and give some examples where some upper bounds of numerical radius are computed and analyzed for certain operators.

In this paper, we aim to develop formulas of spectral radius for an operator in terms of generalized Aluthge transform, numerical radius, iterated generalized Aluthge transform, and asymptotic behavior of powers of . These formulas generalize some of the formulas of spectral radius existing in literature. As an application, these formulas are used to obtain several characterizations of normaloid operators.

Let be any bounded linear operator defined on a complex Hilbert space In this paper, we present some numerical radius inequalities involving the generalized Aluthge transform to attain upper bounds for numerical radius. Numerical computations are carried out for some particular cases of generalized Aluthge transform.