The root operator on invariant subspaces of the Bergman space
type: Item
instance of: scholarly article
title: The root operator on invariant subspaces of the Bergman space
zbMATH Open document ID: 1046.47022
author: Rongwei Yang
author: Kehe Zhu
published in: Illinois Journal of Mathematics
publication date: 2004-01-19T00:00:00Z
review text: Let \(\mathbb D\) be the open unit disc in \(\mathbb C\). Denote by \(dA\) the normalized area measure on \(\mathbb D\), and let \(A^2\) be the usual Bergman space of all analytic functions in \(L^2(dA)\). Its reproducing kernel is \(K(z,w)=(1-z\overline w)^{-2}\). The Bergman shift \(B\) is defined to be the operator of multiplication by \(z\) on \(A^2\). Closed subspaces of \(A^2\) which are invariant for \(B\) are simply called invariant subspaces. Given such a space \(I\), denote by \(T\) the restriction of \(B\) to \(I\) and by \(K^I(z,w)\) its reproducing kernel. The root function of \(I\) is given by \(R^I(z,w)=K^I(z,w)/K(z,w)\), and the associated root operator on \(I\) is defined by \(C^I(f)(z)=\int_\mathbb D R^I(z,w)f(w)\;dA(w)\). It is always true that \(C^I\) defines a bounded operator on \(I\). Among other results, the authors prove that compactness is equivalent to membership in any of the Schatten classes \({\mathcal C}_p\), \(1\leq p<\infty\), and that this can be characterized by finiteness of the so-called index of \(I\) which is, by definition, the codimension of \(zI\) in \(I\). There are also partial results about when the root operator has finite rank.
reviewed by: Hans Jarchow
Mathematics Subject Classification ID: 47B32
Mathematics Subject Classification ID: 47B10
Mathematics Subject Classification ID: 47A15
zbMATH DE Number: 2028869
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2028869