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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.

##### MSC:
 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