Hilbert space general properties

Let/i and/2 be two properties of the species in the system (there may be concentrations, vapor pressures, etc.). If the number of species is finite, one has two vectors fj and f2 in a finite-dimensional vector space (the number of dimensions being the number of species). The inner product is generalized to the inner product in the Hilbert space for two infinite-dimensional vectors fiix),f2(x) as follows (y is the label x when intended as a dummy variable we assume that X has been normalized so as to be dimensionless, see later discussion) ... 

Here, we discuss only a few basic properties of generalized squeezed vacuum, given by (78). By definition, it is properly normalized for arbitrary dimension of the Hilbert space. There are several ways to prove that the generalized squeezed vacuum goes over into the conventional squeezed vacuum ( C)) in the limit of s —> oo. By definition (78), one can conclude that the property lim< x C)(5 = 0)(oo) = 10) holds, since the FD annihilation and creation operators go over into the conventional ones lim< x as = a and lim< x a a. One can also show, at least numerically, that the superposition coefficients (81) approach the coefficients bn for the conventional squeezed vacuum Iims x bn = bn for n = 0,..., s. We apply another method based on the calculation of the scalar product (ClOo) - We show the analytical results for C < 1 only. We have found the scalar product between conventional and generalized squeezed vacuums in the form (for even, v)... 

Now the discussiorr regards the types of terms the projector operators provide for operatorial decomposition on a general Hilbert space (i.e., with an arbitrary vectorial basis) for clearly specifying the cases one has to introduce the notion of invariant subspace Ak9f it exists whenever an associate projector operator Ak exists with the property that transforms into itself under the action of an arbitrary operator ... 

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