Beurling's phenomenon in two variables
The author extends the Beurling's phenomenon from the Hardy space over the unit disk \(H^2(\mathbb{D}^2)\) to the two variable Hardy space \(H^2(\mathbb{D})\). Although in \(H^2(\mathbb{D}^2)\) things are far more complicated and there is no similar characterization of invariant subspaces in terms of inner functions as in \(H^2 (\mathbb{D})\), the author shows in this paper an analogous phenomenon in terms of reproducing kernels. It is well-known that every shift-invariant subspace \(M\) is of the form \(\psi H^2 (\mathbb{D})\) for some inner function \(\psi\) by Beurling's theorem, and the reproducing kernel of \(M\) is \(K_\lambda^M(z)=\frac {\overline {\psi (\lambda)}\psi(z)} {1-\overline\lambda z}\). Since \((1-|\lambda|^2) K_\lambda^M (\lambda)= |\psi(\lambda)|^2\), the fact that \(\psi\) is inner is equivalent to the property that \((1-|\lambda|^2) K_\lambda^M (\lambda)\) has boundary value 1 almost every where on \(\partial\mathbb{D}\), and so \((1-|\lambda|^2) K_\lambda^M(\lambda)\) is subharmonic. For the two variable space \(H^2(\mathbb{D}^2)\), a natural analogue of \((1-|\lambda|^2)K_\lambda(\lambda)\) is \(G^M(\lambda_1, \lambda_2)=(1-|\lambda_1|^2) (1-|\lambda_2|^2)K^M_{(\lambda_1,\lambda_2)} (\lambda_1,\lambda_2)\), where \(K^M_\lambda(z)\) is the reproducing kernel of \(M\). In this paper, the author shows that \(G^M(\lambda_1, \lambda_2)\) is subharmonic in each variable, and under a mild condition it is shown that \(G^M(\lambda_1, \lambda_2)=1\) almost everywhere on \(\partial\mathbb{D}^2\).
scientific article; zbMATH DE number 2101687