A Poincaré-Dulac approach to a nonlinear Beurling-Lax-Halmos theorem
type: Item
instance of: scholarly article
title: A Poincaré-Dulac approach to a nonlinear Beurling-Lax-Halmos theorem
zbMATH Open document ID: 0688.47022
DOI: 10.1016/0022-247X(89)90124-8
author: Joseph A. Ball
author: J. William Helton
author: Allen Tannenbaum
author: Ciprian Foias
published in: Journal of Mathematical Analysis and Applications
publication date: 1989-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 47J05
Mathematics Subject Classification ID: 47A20
Mathematics Subject Classification ID: 46G20
Mathematics Subject Classification ID: 47B35
zbMATH DE Number: 4127937
zbMATH Keywords: Poincaré-Dulac approach to a nonlinear Beurling-Lax-Halmos theorem
zbMATH Keywords: S- invariant subspace
zbMATH Keywords: Hardy space
zbMATH Keywords: Hilbert-Schmidt mapping
zbMATH Keywords: formal power series
reviewed by: Richard M. Aron
MaRDI profile type: Publication
cites work: Infinite Hankel matrices and generalized Caratheodory-Fejer and Riesz problems
cites work: Q4744562
cites work: Q3902017
cites work: Q3797733
cites work: Fading memory and the problem of approximating nonlinear operators with Volterra series
cites work: Q3905961
cites work: Q3244654
cites work: Q3711178
cites work: Q3818918
cites work: Q5600360
cites work: Q5732117
full work available at URL: 0022-247x(89)90124-8
OpenAlex ID: W2169492142
description: scientific article; zbMATH DE number 4127937