H is an observable self adjoint operator, the infinite-dimensional analog of Hermitian matrices.
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A bounded self adjoint operator on Hilbert space is, a fortiori, a bounded operator on a Banach space.
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In quantum mechanics, observables are, not necessarily bounded, self adjoint operators and their spectra are the possible outcomes of measurements.
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The crucial fact here is that, for a bounded self adjoint operator " T " and a polynomial " p ",
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V * is the unique positive square root of " M * M ", as given by the Borel functional calculus for self adjoint operators.
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Via its spectral measures, one can define a decomposition of the spectrum of any self adjoint operator, bounded or otherwise into absolutely continuous, pure point, and singular parts.
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Where J = J ^ {-1 } = J ^ { * } is an antilinear isometry called the modular conjugation and \ Delta = S ^ * S = FS is a positive self adjoint operator called the modular operator.
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However the predual can also be defined without using the Hilbert space that " M " acts on, by defining it to be the space generated by all positive "'normal "'linear functionals on " M " . ( Here " normal " means that it preserves suprema when applied to increasing nets of self adjoint operators; or equivalently to increasing sequences of projections .)
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