Some Banach spaces of vector valued functions and an extremal problem
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 \(\|\Phi(0)c\|_{\mathcal H}\).
scientific article; zbMATH DE number 1355161