H is an observable self adjoint operator, the infinite-dimensional analog of Hermitian matrices.
2.
A bounded self adjoint operator on Hilbert space is, a fortiori, a bounded operator on a Banach space.
3.
In quantum mechanics, observables are, not necessarily bounded, self adjoint operators and their spectra are the possible outcomes of measurements.
4.
The crucial fact here is that, for a bounded self adjoint operator " T " and a polynomial " p ",
5.
V * is the unique positive square root of " M * M ", as given by the Borel functional calculus for self adjoint operators.
6.
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.
7.
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.
8.
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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