On the Carleman formula for a matrix ball
Let \(\mathfrak B\) be a matrix ball and \(\Delta\) is the boundary of the matrix ball. The Hardy space \(\mathcal H^1(\mathfrak B)\) is the space of holomorphic functions satisfying \[ \sup_{0<r<1} \int_\Delta |f(rZ)|d\sigma(Z) < \infty. \] Let \(E\) be a set of a positive measure \(\sigma\) on \(\Delta\). This paper shows the formula, called the Carleman formula, which reconstructs the value of the function \(f\) of the Hardy class \(\mathcal H^1(\mathfrak B)\) in the domain \(\mathfrak B\) by its radial boundary values onto the set \(E\). The author uses the properties of the Bergman, Szegö, and Poisson kernels for \(\mathfrak B\).
scientific article; zbMATH DE number 1541428