Inequalities for the numerical radius, the norm and the maximum of the real part of bounded linear operators in Hilbert spaces (Q924370)

The numerical radius \(w(A)\) of a bounded linear operator \(A\) on a complex Hilbert space \(H\) is the quantity \(w(A)= \sup\{|\langle Ax, x\rangle|: x\in H\), \(\| x\|= 1\}\). It is well-known that \(w(A)\leq\| A\|\leq 2w(A)\). In a previous paper, the author obtained estimates of \(\| A\|- w(A)\), \(w(A)/\| A\|\) and \(\| A\|^2- w(A)^2\) for operators \(A\) satisfying \(\| A-\lambda I\|\leq r\) (\(\lambda\neq 0\), \(r> 0\)) or \(\text{Re}[(A^*-\overline aI)(bI- A)]\geq mI\) (\(a,b\in\mathbb{C}\) and \(m\geq 0\)). In the present paper, he considers \(v_s(A)= \sup\{\text{Re}\langle Ax,x\rangle:\| x\|= 1\}\) and establishes, under similar conditions on \(A\), upper bounds for \(\| A\|- v_s(uA)\), \(w(A)- v_s(uA)\), \(\| A\|^2- v_s(uA)^2\) and \(w(A)^2- v_s(uA)^2\), and lower bounds for \(v_s(uA)/\| A\|\) and \(v_s(uA)/w(A)\) for \(u\) in \(\mathbb{C}\) with \(|u|= 1\). For example, for the operator norm, he has, for \(A\neq 0\) and \(\lambda\neq 0\), (1) \(\| A\|- v_s((\overline\lambda/|\lambda|)A)\leq\| A-\lambda I\|^2/(2|\lambda|)\), and, for \(A\) satisfying \(\| A-\lambda I\|\leq |\lambda|\), (2) \(v_s((\overline\lambda/\lambda|)A)/\| A\|\geq (1- \|(A/\lambda)- I\|^2)^{1/2}\) and (3) \(\| A\|^2- v_s((\overline\lambda/|\lambda|)A)^2\leq 2(|\lambda|-|\lambda|^2-\| A-\lambda I\|^2)^{1/2} v_s((\overline\lambda/|\lambda|)A)\). There are analogous estimates for the numerical radius. Their proofs are all elementary. The paper also considers semi-inner products associated with the operator norm and numerical radius. For example, the superior semi-inner product associated with the norm is given by \(\langle A,B\rangle_{s,n}= \lim_{t\to 0+}(\| B+ tA\|^2-\| B\|^2)/(2t)\) for operators \(A\) and \(B\) on \(H\). They are not used in the proofs of the main results.