Norm Inequalities for Commutators of Normal Operators

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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 1Aa 2, b 1Bb 2, c 1Cc 2, and d 1Dd 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.