Abstract
For A a symmetric and H a self-adjoint (not necessarily semi-bounded) operator on a Hilbert space H, we give conditions in terms of the boundedness of operators of the form (H+z)−p (adH)n(A)(H+z)−q, z∈ℂ, n, p, q ∈ ℕ, which imply essential self-adjointness of A on any core of some power of H. By specializing to the case of semibounded H and/or A, we arrive at the same conclusions under weaker conditions. Our results generalize several previous ones of the same nature. Applications to quantum mechanics and quantum field theory are indicated.
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Driessler, W., Summers, S.J. On commutators and self-adjointness. Lett Math Phys 7, 319–326 (1983). https://doi.org/10.1007/BF00420182
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DOI: https://doi.org/10.1007/BF00420182