Singular inner functions and biinvariant subspaces for dissymmetric weighted shifts (Q674730)

A dissymmetric weight is a non-decreasing unbounded map \(\omega:\mathbb{Z}\to [1,\infty)\) satisfying \(\omega(n)= 1\) for \(n\geq 0\), \(\varlimsup_{n\to-\infty}\omega(n- 1)/\omega(n)< \infty\), and \(\lim_{n\to-\infty} [\omega(n)]^{1/|n|}= 1\). Let \(\Gamma\) be the unit circle and put \[ L^2_\omega(\Gamma)= \Biggl\{f\in L^1(\Gamma): \|f\|_\omega= \biggl(\sum_{n\in\mathbb{Z}} |\widehat f(n)|^2\omega^2(n)\biggr)^{1/2}< \infty\Biggr\}. \] This paper is the first part of a program devoted to the study of biinvariant and hyperinvariant subspaces for the dissymmetric weighted shift \(S_\omega: f(e^{it})\to e^{it}f(e^{it})\), \(f\in L^2_\omega(\Gamma)\). For a singular inner function \(U\) denote by \(E_U\) the closure of \(\text{span}(S^nU)\) in \(L^2_\omega(\Gamma)\). The space \(E_U\) is biinvariant and hyperinvariant for \(S_\omega\). In general case there is always a \(U\) with \(E_U\neq L^2_\omega(\Gamma)\). If \(\omega\) satisfies \(\lim_{n\to\infty}\log \omega(-n)/\sqrt n=\infty\), then for all singular inner \(U\) the subspace \(E_U\) is proper. No non-trivial biinvariant subspaces of \(S_\omega\) were known so far for the quasi-analytic case.