On Norm Inequalities and Orthogonality of Commutators of Derivations

Abstract

Let $H$ be a complex separable Hilbert space and $B(H)$ the algebra of all bounded linear operators on $H$. In this paper, we give considerable generalizations of the inequalities for norms of commutators of normal operators. Let $S, T \in B(H)$ be positive normal operators with the cartesian decomposition $S=A+iC$ and $T=B+iD$ such that $a_{1}\leq A \leq a_{2},\, b_{1}\leq B \leq b_{2},\,c_{1}\leq C \leq c_{2}$ and $ d_{1}\leq D \leq d_{2}$ for some real numbers $a_{1},\,a_{2},\,b_{1},\,b_{2},\,c_{1},\,c_{2},\,d_{1}$ and $d_{2}$ we have shown that $\|ST-TS\|\leq \frac{1}{2}\sqrt{(a_{2}-a_{1})^{2}+(c_{2}-c_{1})^{2}}\sqrt{(b_{2}-b_{1})^{2}+(d_{2}-d_{1})^{2}}.$ Moreover, orthogonality and norm inequalities for commutators of derivation are also established. We have shown that if the pair of operators $(S,T)$ satisfies Fuglede-Putnam's property and $C \in ker(\delta _{S,T})$ where $C\in B(H)$ then $\|\delta _{S,T}X+C\|\geq \|C\|.$