2019/03/25 by Okelo, N. B., Mogotu, P. O.
#47A12 #47A30 #47A63 #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.1903.10358
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 ∈ B(H) be positive normal operators with the cartesian decomposition S=A+iC and T=B+iD such that a1≤ A ≤ a2, b1≤ B ≤ b2, c1≤ C ≤ c2 and d1≤ D ≤ d2 for some real numbers a1, a2, b1, b2, c1, c2, d1 and d2 we have shown that ‖ST-TS‖≤ (1)/(2)√(a2-a1)2+(c2-c1)2√(b2-b1)2+(d2-d1)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 ∈ ker(δS,T) where C∈ B(H) then ‖δS,TX+C‖≥ ‖C‖.