An \(H^{2}\)-corona theorem on the bidisk for infinitely many functions.
The author establishes the \(H^2\)-corona theorem on the bidisk for infinitely many functions. For the general polidisk, an extension of this method applies. Assuming the data are smooth across the boundary, the author explicitly writes a solution for the \(H^2\) problem, using the representation of the kernel of a matrix valued analytic function. Estimates to show that this solution has the appropriate size are based on iterating the one-variable Littlewood-Paley identity. Let \(D\) be the open unit disk. \(F({\mathbf z})\) denotes the \(n\times m\) matrix, giving a linear transformation in \(B(l^2_m,l^2_n)\) in the natural way. Assume that \(\varepsilon\leq F({\mathbf z})F({\mathbf z})^*\leq 1\) for all \({\mathbf z}\in D^2\). Then \[ T_FT_F^*\geq\left(731e\left[ \frac{1}{\varepsilon^5}\ln \frac{1}{\varepsilon^2} \right]\right)^{-2}\text{I}, \] where \(T_F\) is the analytic Toeplitz operator received by restricting \(F\) to \(H^2(D^2)^{(m)}\).
scientific article; zbMATH DE number 2056685