Inequalities for some functionals associated with bounded linear operators in Hilbert spaces (Q945588)

For a bounded linear operator \(A\) on a real or complex Hilbert space \(H\) with inner product \(\langle\cdot,\cdot\rangle\), let \(w(A)= \sup\{|\langle Ax,x\rangle|: x\in H,\| x\|= 1\}\) and \(m(A)= \{|\langle Ax,x\rangle|: x\in H,\| x\|= 1\}\). Other similar functionals with \(\text{Re}\langle Ax,x\rangle\) and \(\text{Im}\langle Ax,x\rangle\) replacing \(|\langle Ax,x\rangle|\) in the above expressions can also be defined. Moreover, these can even be extended to two operators \(A\) and \(B\) on \(H: w_e(A, B)= \sup\{(|\langle Ax,x\rangle|^2+ |\langle Bx,x\rangle|^2)^{1/2}: x\in H,\| x\|= 1\}\) and \(m_e(A, B)\) with supremum above replaced by the infimum. In this paper, various inequalities relating these quantities and the operator norms are presented. For example, it is shown that, for any two scalars \(a\) and \(b\), the inequalities \[ {1\over 4}|a- b|^2\leq m\Biggl(A- {a+b\over 2} I\Biggr)^2+ \begin{cases} {1\over 2} w_e(aI- A,A- bI)^2,\\ w(aI- A)w(A- bI),\end{cases} \] and \[ \| A\|^2- w(A)^2\leq \Biggl\| A-{a+ b\over 2} I\Biggr\|^2- m\Biggl(A-{a+ b\over 2} I\Biggr)^2 \] hold. The proofs for these are all elementary.