Some Banach spaces of vector valued functions and an extremal problem
type: Item
instance of: scholarly article
title: Some Banach spaces of vector valued functions and an extremal problem
zbMATH Open document ID: 0937.46029
DOI: 10.1007/BF02789838
author: R. Cheng
published in: Analysis Mathematica
publication date: 1999-10-31T00:00:00Z
review text: This paper is concerned with an extension to vector-valued functions of a classical result of Szegő who had determined the infimum of \(\int_{0}^{2\pi} |f(e^{i\theta})|^{p} w(e^{i\theta}) d\theta/2\pi \) over all functions in the Hardy space \(H^{p}\) subject to the condition \(f(0)=1\), where \(w\) is a positive weight function and \(1<p<\infty\). The set-up here is as follows: For a Hilbert space \(\mathcal H\) let \(W\) be an \(L^{\infty}\)-function on the unit circle whose values are positive bounded operators on \(\mathcal H\). The space \(L^{p}(W)\) is the completion of the space of \(\mathcal H\)-valued trigonometric polynomials for the norm \[ \|f\|_{p,W}= ( \int_{0}^{2\pi} \|W(e^{i\theta})^{1/p} f(e^{i\theta}) \|_{\mathcal H}^{p} d\theta/2\pi)^{1/p}. \] The extremal problem considered here is to find the infimum of \(\|c+c'f\|_{p,W}\) over all \(c'\in \mathcal H\) and \(f\in H^{p}\) with \(f(0)=0\). The solution involves a decomposition of \(W=(\Phi^{*}\Phi)^{p/2} + W_{\infty}\) for some outer function \(\Phi\) along with a Wold type decomposition. The looked-for infimum is then \(\|\Phi(0)c\|_{\mathcal H}\).
Mathematics Subject Classification ID: 46E40
zbMATH DE Number: 1355161
zbMATH Keywords: vector-valued functions
zbMATH Keywords: Hardy space
zbMATH Keywords: Wold decomposition
zbMATH Keywords: extremal problem
reviewed by: Dirk Werner
MaRDI profile type: Publication
cites work: Innovations and Wold decompositions of stable sequences
cites work: A matrix generalization of a theorem of Szegö
cites work: Wold decomposition, prediction and parameterization of stationary processes with infinite variance
cites work: Q3711178
cites work: Q5595759
description: scientific article; zbMATH DE number 1355161