Rudin orthogonality problem on the Bergman space
Let \(dA\) denote the area measure in the complex plane \(\mathbb{C}\), normalized so that the measure of the open unit disk equals 1. The Bergman space \(L^2_a(D)\) is the Hilbert space of holomorphic functions on \(D\) that are also square integrable functions on \(D\). The authors prove the following theorem: Let \(\phi\) be analytic on a neighborhood of the closed unit disk. If \(\{\phi^k: k= 0,1,\dots\}\) forms an orthogonal set in \(L^2_a(D)\), then \(\phi(z)= cz^n\) for an integer \(n\) and some constant \(c\). The authors conjecture that the above statement is true when \(\phi\) is bounded and analytic in \(D\). They also obtain as a consequence of their result that an analytic multiplication operator on the Bergman space is unitarily equivalent to a weighted unilateral shift of finite multiplicity \(n\) if and only if its symbol is a constant multiple of the \(n\)th power of a Möbius transform. This latter result was obtained previously by the theory of the Hardy space of the bi-disk.
scientific article; zbMATH DE number 5898826
scientific article