Norm Inequalities for Commutators of Normal Operators

Abstract

Let S, T, and X be bounded linear operators on a Hilbert space. It is shown that if S and T are normal with the Cartesian decompositions S = A+iC and T = B+iD such that a ₁ ≤ A ≤ a ₂, b ₁ ≤ B ≤ b ₂, c ₁ ≤ C ≤ c ₂, and d ₁ ≤ D ≤ d ₂ for some real numbers a ₁, a ₂, b ₁, b ₂, c ₁, c ₂, d ₁, and d ₂, then for every unitarily invariant norm |||·|||,

and

$$ \left\| {ST - TS} \right\| \leqslant \frac{1} {2}\sqrt {(a_2- a_1 )^2+ (c_2- c_1 )^2 } \sqrt {(b_2- b_1 )^2+ (d_2- d_1 )^2 } , $$

where ‖·‖ is the usual operator norm. Applications of these norm inequalities are given, and generalizations of these inequalities to a larger class of nonnormal operators are also obtained.