New Reverse Inequalities for the Numerical Radius of Normal Operators in Hilbert Spaces

Let (H ; 〈·, ·〉) be a complex Hilbert space and T : H → H a bounded linear operator on H. Recall that T is a normal operator if T T = TT . Normal operator T may be regarded as a generalisation of self-adjoint operator T in which T ∗ need not be exactly T but commutes with T [5, p. 15]. An equivalent condition with normality that will be extensively used in the following is that ‖Tx‖ = ‖T ∗x‖ for any x ∈ H. The numerical range of an operator T is the subset of the complex numbers C given by [5, p. 1]: W (T ) = {〈Tx, x〉 , x ∈ H, ‖x‖ = 1} .