Numerical radius ($w(T)={\sup }_{\Vert u\Vert =1}|⟨Tu,u⟩|$)

For a bounded linear operator on a complex Hilbert space, the numerical radius is the supremum of the absolute values in its numerical range of an operator. It is a norm equivalent to the operator norm, with $w(T)\le \Vert T\Vert \le 2w(T)$. For a self-adjoint operator the two norms agree.