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A survey of some recent inequalities for the norm and numerical radius of operators in Hilbert spaces. (English) Zbl 1136.47006
This is a survey on some inequalities involving norms and numerical radii of (one or two) bounded linear operators on a complex Hilbert space obtained in recent years by the author. Proofs are also provided.
After the preliminaries in Section 1, estimates for the difference \(\| T\|- w(T)\) or the ratio \(w(T)/\| T\|\) for an operator \(T\) are given in Section 2. An example of the former is \(\| T\|- w(T)\leq r^2(2|\lambda|)\) for \(T\) satisfying \(\| T-\lambda I\|\leq r\), where \(\lambda\neq 0\) and \(r>0\), and of the latter \(w(T)/\| T\|\geq (1-(r^2/|\lambda|^2))^{1/2}\) for \(\| T-\lambda I\|\leq r\), where \(|\lambda|>r>0\).
In Section 3, the author considers the inequalities \[ w(T)^2\leq (w(T)^2)+\| T\|^2)/2 \] and \[ \| T\|^2-w(T)^2\leq\text{inf}_{t\in \mathbb{R}}\{\| T-tI\|^2, \| T-itI\|^2\}. \] Section 4 gives inequalities involving two operators \(A\) and \(B\) on a space \(H\). For example, it is shown that \(\|(A*A+ B*B)/2\|\leq w(R* A)+ (r^2/2)\) if \(\| A\|\leq r\) \((r> 0)\). Finally, in Section 5, inequalities are given for the case when \(B\) is invertible, such as \(\| A\|\leq\| B^{-1}\|(w(B*A)+ (r^2/2))\) if \(\| A-B\|\leq r\) \((r> 0)\) and \(B\) is invertible.
The proofs of these inequalities are all elementary, involving only basic properties of the norms and numerical radii of operators.
We remark that in the statement of Theorem 1.7, “\(w(T)=\| T\|/2\)” should be replaced by “\(w(T)=\| T\|/2= 1/2\)” in order for the conclusion to hold.

47A30 Norms (inequalities, more than one norm, etc.) of linear operators
47A12 Numerical range, numerical radius
47-02 Research exposition (monographs, survey articles) pertaining to operator theory