The root operator on invariant subspaces of the Bergman space
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.
scientific article; zbMATH DE number 2028869