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Simple spectrum of the tensor product of powers of a mixing automorphism. (English. Russian original) Zbl 1275.37008
Trans. Mosc. Math. Soc. 2012, 183-191 (2012); translation from Tr. Mosk. Mat. O.-va 73, No. 2, 229-239 (2012).
The main result of the paper is the construction of a mixing automorphism \(T\) such that the tensor product \(T\otimes T^2\otimes T^3\otimes\cdots\) has simple spectrum. Here, by “mixing” it is meant the convergence \(T^n\to P\), where \(P\) is the orthoprojection onto the space of constants in \(L_2(X,\mu)\). To prove the main result, the author provides constructions of rank-1 transformations with mixing property. Further, the simplicity of the spectrum of tensor product of powers of special nonmixing transformations is proved. Then, preserving the spectral property, the required assertion is proved.

MSC:
37A30 Ergodic theorems, spectral theory, Markov operators
28D05 Measure-preserving transformations
47A35 Ergodic theory of linear operators
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