Journal of Operator Theory

Volume 93, Issue 2, Spring 2025 pp. 393-412.

The Howland-Kato commutator problem. II

Authors: Richard Froese (1), Ira Herbst (2)
Author institution: (1) Department of Mathematics, University of British Columbia, Vancouver, Canada
(2) Department of Mathematics, University of Virginia, Charlottesville, VA, U.S.A.

Summary: We continue the search, begun by T. Kato, for all pairs of real and bounded measurable functions $\{f,g\}$ on the line that result in a positive commutator $\mathrm i[f(P),g(Q)]$. Here $P$ and $Q$ are the usual Heisenberg operators: $Q$ is multiplication by the coordinate $x$ in $L^2(\mathbb{R},\mathrm dx)$ and $P = -\mathrm i\,\mathrm d/\mathrm dx$ on the appropriate domains. We prove a number of partial results including a connection with Loewner's celebrated theorem on matrix monotone functions.

DOI: http://dx.doi.org/10.7900/jot.2023may20.2470
Keywords: commutator, Heisenberg operators, positivity

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