Approximation of the Bergman norm by the norms of the direct product of two Szegő spaces

Abstract:

Let G be a bounded regular region in the plane. Let $H_2^{1/2}(G)$ denote the Szegő space of G composed of analytic functions on G with finite norm \[ {\left \{ {\frac {1}{{2\pi }}\int _{\partial G} {|f(z){|^2}|dz|} } \right \}^{1/2}} < \infty .\] We set $f(z) = \Sigma _{j = 1}^\infty {\varphi _j}(z){\psi _j}(z)\;({\varphi _j},{\psi _j} \in H_2^{1/2}(G))$. Then, we determine a necessary and sufficient condition for $f(z)$ to make the equality \[ \frac {1}{\pi }\int {\int \limits _G {|f(z){|^2}\;dx\;dy = \min \left \{ {\sum \limits _{j = 1}^\infty {\sum \limits _{k = 1}^\infty {\frac {1}{{2\pi }}\int _{\partial G} {{\varphi _j}({z_1})\overline {{\varphi _k}({z_1})} |d{z_1}|\frac {1}{{2\pi }}\int _{\partial G} {{\psi _j}({z_2}} )\overline {{\psi _k}({z_2})} |d{z_2}|} } } } \right \}} } \] hold. The minimum is taken here over all analytic functions $\Sigma _{j = 1}^\infty {\varphi _j}({z_1}){\psi _j}({z_2})$ on $G \times G$ satisfying $f(z) = \Sigma _{j = 1}^\infty {\varphi _j}(z){\psi _j}(z)$ on G.