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A counterexample to the Howland-Kato problem for positive commutators

The Howland-Kato conjecture that every nonzero positive commutator $i[f(P),g(Q)]$ must arise from functions in appropriate Kato classes is false: $i[\arctan(P),\arctan(Q)]$ is nonzero and nonnegative.

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analysisAug 7, 2026Significance 25/100Registry: unreviewed

A counterexample to the Howland-Kato problem for positive commutators

Prior state unknown→disproved

The Howland-Kato conjecture that every nonzero positive commutator $i[f(P),g(Q)]$ must arise from functions in appropriate Kato classes is false: $i[\arctan(P),\arctan(Q)]$ is nonzero and nonnegative.

Source authenticatedA counterexample to the Howland-Kato problem for positive commutators →
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The Howland-Kato conjecture that every nonzero positive commutator $i[f(P),g(Q)]$ must arise from functions in appropriate Kato classes is false: $i[\arctan(P),\arctan(Q)]$ is nonzero and nonnegative.

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