The commutator i[arctan(P), arctan(Q)] is nonnegative with trace pi/2, disproving the Kato conjecture.
Simon,Loewner’s theorem on monotone matrix functions, Grundlehren Math
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A counterexample to the Kato conjecture for positive commutators
The commutator i[arctan(P), arctan(Q)] is nonnegative with trace pi/2, disproving the Kato conjecture.