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Spectrum, resolvent, numerical range, functional calculus, operator semigroups. Special classes of operators: compact, Fredholm, dissipative, differential, integral, pseudodifferential, etc.

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Self-adjointness of an operator

This is a problem from page 3 of Bourgain, Burq, and Zworski - Control for Schrödinger equations on 2-tori: rough potentials, the author claim that Hence $P = - \partial_x^2 + W$ defined on $C^\infty …