Let G be a nontrivial torsion-free group and has a solution over G provided a single relation on coefficients of s(t) holds. We also generalize our results to equations containing higher powers of t. The later equations are also related to Kaplansky zero-divisor conjecture.

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.

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.

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.

Levin conjecture states that every group equation is solvable over any torsion free group. The conjecture is shown to hold true for group equation of length seven using weight test and curvature distribution method. Recently, these methods are used to show that Levin conjecture is true for some group equations of length eight and nine modulo some exceptional cases. In this paper, we show that Levin conjecture holds true for a group equation of length nine modulo 2 exceptional cases. In addition, we present the list of cases that are still open for two more equations of length nine.