Strongly mixing operators on Hilbert spaces and speed of mixing (Q2840183)

The author investigates the speed of mixing for operators on infinite-dimensional (complex) Hilbert spaces which are strongly mixing with respect to a non-degenerate Gaussian measure. He shows that there is no universal rate of mixing for all \(L^2\)-functions. However, for sufficiently regular functions, the author obtains a speed \(n^ {-\alpha}\), provided the operator satisfies another regularity property, namely the eigenvectors associated to the unimodular eigenvalues of the operator are parametrized by an \(\alpha\)-Hölder eigenvector field. Recall that strongly mixing operators are ergodic. NEWLINENEWLINENEWLINEThe first section introduces the basic definitions. The second section recalls known results which will be essential for the rest of the paper. Recall that for a non-degenerate Gaussian measure \(m\) on a (complex) Hilbert space \(\mathcal{H}\), the map \(\langle x,\cdot\rangle:y\mapsto \langle x,y\rangle\) has a complex symmetric Gaussian distribution for all \(x\in\mathcal{H}\) and assigns positive measures to non-empty open sets. A bounded operator \(T\in\mathcal{B}(\mathcal{H})\) is strongly mixing relative to an invariant measure if NEWLINE\[NEWLINE \int f(T^n(x))g(x)\, \operatorname{d}m(x)\longrightarrow\int f\operatorname{d}m\int g\,\operatorname{d}m, NEWLINE\]NEWLINE for all \(f,g\in L^2(\mathcal{H},m)\) as \(n\) goes to \(\infty\). The speed of mixing is the rate of convergence to zero of the sequence of correlations NEWLINE\[NEWLINE \mathcal{I}_n(f,g)=\|\int f(T^n(x))g(x)\,\operatorname{d}m(x)-\int f\operatorname{d}m\int g\,\operatorname{d}m\|\,NEWLINE\]NEWLINE A bounded operator \(T\in\mathcal{B}(\mathcal{H})\) is \(\mu\)-spanning if the eigenspaces associated to \(\mu\)-a.e. unimodular eigenvalues span a dense subset of \(\mathcal{H}\), where \(\mu\) is the normalized Lebesgue measure on \(S^1\). This kind of operators admits a non-degenerate invariant Gaussian measure, and \(T\) is strongly mixing relative to this measure; see [\textit{F. Bayart}and \textit{S. Grivaux}, Trans. Am. Math. Soc. 358, No. 11, 5083--5117 (2006; Zbl 1115.47005)]). NEWLINENEWLINENEWLINE An eigenvector field is a measurable map \(E:S^1\to\mathcal{H}\) mapping \(\lambda\) to a \(\lambda\)-eigenvector (or \(0\)). Through the paper the existence of a unique eigenvector field is supposed; the author reveals, in Section 2 and 3 and in the final comments of the paper, a guide to avoid this limitation. Supposing \(E\) is \(\alpha\)-Hölder, the author shows that, in the special cases where \(f=\langle x,\cdot\rangle\) and \(g=\overline{\langle y,\cdot\rangle}\), the correlations \(\mathcal{I}_n(f,g)\) go to zero with speed \(n^{-\alpha}\). This section finishes with a remarkable family of examples (Theorem 2.16) where the speed of mixing is exactly \(n^{-\alpha}\) for every \(\alpha\in (0,1)\).NEWLINENEWLINESection 3 shows a negative result (Theorem 3.4): There is no uniform speed of mixing for all the functions in \(L^2(\mathcal{H},m)\) for all \(\mu\)-spanning bounded operators. This theorem is, in fact, a direct consequence of a result due to \textit{C. Badea} and \textit{V. Müller} [Topology Appl. 156, No. 7, 1381--1385 (2009; Zbl 1204.47001)], where it is shown that for every bounded operator \(S\) on a complex Hilbert spaces with spectral radius equal to \(1\) and for any sequence \((s_n)\) converging to 0, there exists an element \(x\) such that \(\langle S^n x,x\rangle\geq s_n\) for all \(n\in\mathbb{N}\).NEWLINENEWLINENEWLINEThe final sections are much more technical. Here, the author restricts to real-valued \(L^2(\mathcal{H},m)\) maps. Fock decomposition and Wick transformations [\textit{V. Peller}, Hankel operators and their applications. New York, NY: Springer (2003; Zbl 1030.47002)] are the tools allowing the author to restrict the study of the speed of mixing to homogeneous polynomials on the functions \(\mathrm{Re}\langle e_k,\cdot\rangle\) (\(\mathrm{Re}\) is the ``real part'' operator and \(\{e_k\}\) a suitable orthonormal basis). Using the estimates of the speed of mixing for this kind of functions, the author succeeds in obtaining regularity conditions on \(f\) and \(g\) in order to have speed of mixing at least \(n^{-\alpha}\) (assuming the existence of an \(\alpha\)-Hölder eigenvector field). These regularity conditions are too technical to be included here. However, it should be noticed that this class of regular functions contains the infinitely differentiable functions such that the square norm of multilinear forms given by the integral representation of the \(k\)-differential has a growth less than or equal to exponential. It is also noticed that the existence of an analytic eigenvector field leads to an exponential speed of mixing.