Inequalities for the numerical radius, the norm and the maximum of the real part of bounded linear operators in Hilbert spaces
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.
maximum and minimum of the real part of bounded linear operators
scientific article; zbMATH DE number 5275779
scientific article