A Poincaré-Dulac approach to a nonlinear Beurling-Lax-Halmos theorem
The authors continue their study of nonlinear versions of results from operator theory [see, e.g. Indiana J. Math. 36, 693-709 (1987; Zbl 0652.47004)]. The theorem which is being extended here is the one stated in the title: let S be a shift on Hilbert space H and let K be a closed, S-invariant subspace of H. Then there is a bounded linear operator \(\theta\) from a Hardy space \(H_ 0\), equipped with the standard shift \(S_ 0\) (i.e. multiplication by exp(it)), to H such that \(K=\theta H_ 0\) and \(\theta S_ 0=S\theta\). Roughly speaking, the authors prove a local, holomorphic version of this result for which the latter equality is true to arbitrary order n. In somewhat more precise terms, let M be a submanifold of H which is locally analytic near 0. Thus, there is a surjective analytic map \(\Phi\) taking an open O-neighborhood \(B_ 0\) in some Hilbert space onto \(B\cap M\) for some open O-neighborhood B in H. Suppose that M is S-invariant; that is, Sx\(\in B\cap M\) whenever \(x\in B\cap M\) and Sx\(\in B\). Assume that the nonlinear part of \(\Phi\) has a factorization \({\hat \Phi}\)W, where \({\hat \Phi}\) is holomorphic and W is a linear Hilbert-Schmidt mapping, and that the domain \(B_ 0\) of the parametrization \(\Phi\) of M is a O-neighborhood in a Hardy space \(H_ 0\). Then there is a formal power series from \(H_ 0\) to \(H_ 0\), \(h_ 0+\sigma_ 2(h_ 0)+\sigma_ 3(h_ 0)+...\), which satisfies the following conditions. For every \(n\geq 2\) and every function \(\sigma\) which is analytic near 0 in \(H_ 0\) and which is such that \(\| \sigma (h_ 0)\| =O(\| h_ 0\|^{n+1}),\) the function \(\Phi^{(n)}:\equiv \Phi (h_ 0+\sigma_ 2(h_ 0)+...+\sigma_ n(h_ 0)+\sigma (h_ 0))\) satisfies \(\| S\Phi^{(n)}(h_ 0)- \Phi^{(n)}(S_ 0h_ 0)\| =O(\| h_ 0\|^{n+1}).\) The authors provide several examples and possible applications.
Poincaré-Dulac approach to a nonlinear Beurling-Lax-Halmos theorem
scientific article; zbMATH DE number 4127937
scientific article