Abstract
Upper and lower bounds are found for the smallest constants αp and βp for which the following inequalities hold: ∥|A|B−B|A|∥p⩽αp∥AB−BA∥p, ∥|A|−|B|∥p⩽βp∥A−B∥p for all self-adjoint operators A and B on a Hilbert space.
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