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If (Formula presented) is an operator on a Banach space (Formula presented) in (Formula presented) is an analytic vector of (Formula presented) if the series expansion of (Formula presented) has a positive radius of absolute convergence; i.e., if (Formula presented) for some finite (Formula presented) See, for example, E. Nelson, Ann. Math. 70, 572 (1959);M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. I: Functional Analysis (Academic, New York, 1972), p. 201.
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14
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85037204670
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An alternating series (Formula presented) in which each (Formula presented) is positive, converges if (1) (Formula presented) for every value of n and (2) (Formula presented)
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An alternating series (Formula presented) in which each (Formula presented) is positive, converges if (1) (Formula presented) for every value of n and (2) (Formula presented)
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15
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85037244050
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An operator (Formula presented) is self-adjoint if it is Hermitian and (Formula presented) where (Formula presented) and (Formula presented) denote the dense domains of the operators (Formula presented) and (Formula presented) respectively. Only self-adjoint operators may be exponentiated to give unitary operators. For precise definitions and details, see M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. I: Functional Analysis (Ref
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An operator (Formula presented) is self-adjoint if it is Hermitian and (Formula presented) where (Formula presented) and (Formula presented) denote the dense domains of the operators (Formula presented) and (Formula presented) respectively. Only self-adjoint operators may be exponentiated to give unitary operators. For precise definitions and details, see M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. I: Functional Analysis (Ref. 13), p. 255.
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