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On the theory of normal dynamic systems. (English. Russian original) Zbl 0197.39401

Sov. Math., Dokl. 3, 625-628 (1962); translation from Dokl. Akad. Nauk SSSR 144, 9-12 (1962).
From the text: Let \(s\) be the space of real infinite sequences \(\{\omega_k\}_{k=-\infty}^\infty\) with the weak topology. Let \(T\) be the shift operator on \(s\), \(T\{\omega_k\} = \{\omega_{k+1}\}\), \(\mu\) a Gaussian measure on \(s\). \(\{s,\mu,T\}\) is called a one-dimensional normal dynamic system (n.d.s.) if the following conditions hold: \[ \int_s \omega_k \,d\mu = 0,\quad \int_s \omega_k^2 \,d\mu = 1,\quad k = 0, \pm1,\ldots; \quad \mu(TA) = \mu(A) \] where \(A\) is a Borel set in \(s\). There exists a symmetric measure \(F(d\lambda)\) on \([-\pi, \pi]\) called the spectral measure of \(\{s,\mu,T\}\). To each symmetric measure \(G(d\lambda)\) on \([-\pi, \pi]\) there is defined the unitary ring \(L_G = \oplus_{n=0}^\infty L_{G^{(n)}}^2\), where \(G^{(n)}\) is the product measure.
Theorem: Let \(\{s,\mu,T\}\) be a one-dimensional n.d.s. with spectral measure \(F(d\lambda)\). The automorphism \(U_T\) of the unitary ring \(L_\mu^2(s)\) defined by \((U_TX)(\omega) =X(T\omega)\) is isomorphic to the automorphism \(V\) on \(L_T\) defined by \[ (Vf_n) (\lambda_1,\ldots,\lambda_n) = \exp [i(\lambda_1 +\ldots + \lambda_n)] f_n(\lambda_1,\ldots,\lambda_n). \] This result is generalized to \(n\)-dimensional n.d.s. by introducing normal automorphisms \[ V_\varphi: (V_\varphi f_n) (\lambda_1,\ldots,\lambda_n) = \exp [i\sum \varphi(\lambda_j)] f_n(\lambda_1,\ldots,\lambda_n). \]

MSC:

47-XX Operator theory
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