Inequalities for Some Functionals Associated with Bounded Linear Operators in Hilbert Spaces

Abstract

Some inequalities between the operator norm, numerical radius and the functionals $v_p$, ${\delta }_p$ defined in terms of the real and imaginary part of $⟨Ax,x⟩$, $x\in H$, $\Vert x\Vert =1$ are established. New upper bounds for the nonnegative quantity $\Vert A{\Vert }^2-w^2(A)$ with $A\in B(H)$ that complement some recent results of the author are given as well.