On Norm Inequalities and Orthogonality of Commutators of Derivations

N. B. Okelo; P. O. Mogotu

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On Norm Inequalities and Orthogonality of Commutators of Derivations

N. B. Okelo, P. O. Mogotu

Functional Analysis · Submitted 25 Mar 2019

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim 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.

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arXiv listing abstract v1. Results and the main claim are underlined; show every role.

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 $\|\delta _{S,T}X+C\|\geq \|C\|.$

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Let $H$ be a complex separable Hilbert space and $B(H)$ the algebra of all bounded linear operators on $H$.	background 1.00	0.05	0.01	asserted 0.75
s1	In this paper, we give considerable generalizations of the inequalities for norms of commutators of normal operators.	contribution 1.00	0.89	0.13	asserted 0.97
s2	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.	result 0.99	0.94	0.57	asserted 1.00
s3	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 $\\|\delta _{S,T}X+C\\|\geq \\|C\\|.$	result 0.99	0.93	0.15	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Original 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 $\|\delta _{S,T}X+C\|\geq \|C\|.$

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