Abstract:
New norm inequalities for commutators of Hilbert space operators are given. Among other inequalities, it is shown that if $A,$$ B$$\in \mathbb{B(H)}$ and there exists a real number $z_{0}$, such that $ \Vert A-z_{0}I\Vert=D_{A} $, then \begin{eqnarray*} \Vert AB \pm BA^*\Vert \leq 2 D_{A} \Vert B \Vert, \end{eqnarray*} where ${{D}_{A}}=\underset{\lambda \in \mathbb{C}}{\mathop{\inf }} \left\| A-\lambda I \right\| $. In particular, under some conditions, we prove that \begin{eqnarray*} \Vert AB\Vert \leq D_{A} \Vert B \Vert, \end{eqnarray*} which is an improvement of submultiplicative norm inequality. Also, we prove several numerical radius inequalities for products of two Hilbert space operators.
Keywords:bounded linear operator, Hilbert space, norm inequality, numerical radius.
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