Toeplitz operators with vertical symbols acting on the poly-Bergman spaces of the upper half-plane
Recall that the \(n\)-analytic Bergman space \(\mathcal{A}^2_n(\Pi)\) on the upper half-plane \(\Pi\) is the closed subspace of \(L_2(\Pi)\) that consists of all \(n\)-analytic functions, i.e., those functions \(\varphi\) that satisfy the equation \((\partial/\partial \overline{z})^n \varphi = 0\). The authors study the \(C^*\)-algebra generated by all Toeplitz operators \(T_a\) acting on \(\mathcal{A}^2_n(\Pi)\) whose bounded measurable symbols \(a\) depend only on \(y =\mathrm{Im}\,z\) and have limit values at \(y=0\) and \(y = \infty\). They show the the above \(C^*\)-algebra is isometrically isomorphic to the algebra of all \(n\times n\) matrix-functions that are continuous on the positive real axis \(\mathbb{R}_+\) and are the scalar multiples of identity at \(y=0\) and \(y = \infty\).
scientific article; zbMATH DE number 6506176
scientific article