The current two main methods for computing the numerical are level-set approach of Mengi and Overton cutting-plane method Uhlig, but until now, no formal convergence rate results have been established either. In practice, which is faster also unclear, with Uhlig's sometimes being much than Overton's approach, on other problems, slower. this paper, we clarify issue propose three improved methods...

Let a be positive element in unital $$C^*$$ -algebra $$\mathfrak {A}$$ . We define semi-norm on , which generalizes the a-operator and a-numerical radius. investigate basic properties of this prove inequalities involving it. Further, we derive new upper lower bounds for radii elements Some other related results are also discussed.

Let F (A) be the numerical range or the numerical radius of a square matrix A. Denote by A◦B the Schur product of two matrices A and B. Characterizations are given for mappings on square matrices satisfying F (A ◦ B) = F (φ(A) ◦ φ(B)) for all matrices A and B. Analogous results are obtained for mappings on Hermitian matrices. 2000 Mathematics Subject Classification. 15A04, 15A18, 15A60

We give several sharp inequalities for the numerical radius of Hilbert space operators .It is shown that if A and B are bounded linear operators on complex Hilbert space H , then 1 2 1 2(1 ) 2(1 ) 2 2 2 2 1 ( ) 2 ( ) 2 r r r r r r w A B A B A B A B α α α α − − − ∗ ∗ ⎛ ⎞ + ≤ + + + + + ⎜ ⎟ ⎝ ⎠ , for 0<r 1 ≤ and ( ) 1 , 0 ∈ α , and if ( ) n A M ∈ , then 2 1 ( ) 4 w A ≤ ( ) 2 2 A A A A ∗ ∗ + + − , ...