Spectral analysis of certain compact factors for Gaussian dynamical systems (Q1360001)

Let \(T\) be a Gaussian ergodic automorphism on a Borel space \((X,\mathbf B,\mu)\). Among spectral measures \(\sigma_f\) defined by \(\{f\circ T^n\}\) on \(L^2(X,\mathbf B,\mu)\) there is a dominating one, denoted by \(\sigma_T\), called the maximal spectral type of \(T\). Then the action of \(T\) on the complex space \(L^2(X,\mathbf B,\mu)\) is determined by \(\sigma_T\) and \(M_T\) -- the multiplicity function of \(T\) -- a function defined on the unit circle and valued on \(\{1,2,\cdots,+\infty\}\). The essential supremum of \(M_T\) is called the maximal spectral multiplicity. Let the homogeneous chaos decomposition of \(L^2(X,\mathbf B,\mu)\) be \(\bigoplus^\infty_{n=0}\mathcal H_c^{(n)}\). A factor of \(T\) is a \(T\)-invariant \(\sigma\)-algebra \({\mathbf A}\). Let \(C(T)=\{S:(X,\mathbf B,\mu)\to(X,\mathbf B,\mu):STS^{-1}=T\}\) and \(\mathcal P\) be a compact subgroup of \(C(T\mid\mathcal H_c^{(1)})\). The factor \(\mathbf A(\mathcal P)\) is defined by the set of Borel sets invariant under all \(S\) in \(\mathcal P\). The main result in the present paper is: If the spectrum of \(T\) on \(\mathbf A(\mathcal P)\) is not simple then its maximal spectral multiplicity on \(L^2(\mathbf A(\mathcal P))\) is equal to \(\infty\). Moreover, the multiplicities in all \(L^p(\mathbf A(\mathcal P),\mu)\) are the same for all \(1<p<+\infty\). Research papers related to this study are [\textit{A. Iwanik}, Lect. Notes Math. 1545, 124-127 (1992; Zbl 0763.58017)] and [\textit{A. Iwanik} and \textit{J. de Sam Lazaro}, C. R. Acad. Sci., Paris, Sér. I 312, No. 11, 875-876 (1991; Zbl 0748.60035)].