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.

this paper deals with the threshold radius that warrants the use of single restraining rail for transit and railroads. different railways use different threshold radius from 91m to 365m to provide restraining rail. these threshold figures are probably experience driven choices. the need for restraining rail may be related to a so-called derailment alarm coefficient of 1.2 sustained over 2m leng...

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 ∗ ∗ + + − , ...

Some power inequalities for the numerical radius based on recent Dragomir extension of Furuta's inequality are established. particular cases also provided. Moreover, we get an improvement H\"older-McCarthy operator in case when $r\geq 1$ and refine generalized involving powers sums products Hilbert space operators.