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By Ishiguro M., Sakamoto Y.

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Example text

2) we obtain (QG a, QGb) k = (G a, Gb) d 9 However, ~Ga(') , a E Rd~ generates L2(R d) and therefore Q : L2(R d) ~ H k is an isometric isomorphism between these two spaces. 10). 6) (F,G) k X = Wxd(f) , . 7) with Proof. q. ~f the real valued then ~ E Hk . 8) fd~. g. gives E(X(a) X) I Rd - d,1 -~ I ~ (b) db Ibl 1/2 where ~(b) (~d+1) = E L2(Rd; C) = T r 1 6 2H k . 1 Let ~ . Since ;(b) 2 d ; C) is in Lo(R ~,~ r C~o(Rd) = ~(-b) and and therefore be real valued and such that 48 ~(0) = ~(0) = 0 . ,a n) .

G, g) , p is the Lev2 metric and (Xk) i_~s dense sequence in the unit sphere of H , ~ complete. weak. convergence . . of. the. sequences . )x, x) weakly~ m x for x's from a dense subset of H is sufficient.

Spectral measures correspond to self-adjoint operators (physical quantities, observables). Semispectral measures may be in the natural way interpreted in quantum statistics (in the theory of decision functions) as randomized strategies. More precisely, randomized strategies are described by commutative semispectral measures (see [2]). 148). In ~ I we shall consider the convergence of observables (self-adjoint operators) in terms of semispectral measures theory. As the starting point to our discussion we shall use considerations to be found in [3].

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