Isometrical identities for the Bergman and the Szegö spaces on a sector (Q804740)

Let \(\Delta (\alpha)=\{z:| \arg z| <\alpha \}\) and consider the Bergman space \[ B_{\Delta (\alpha)}=\{F;\;F\text{ is analytic on } \Delta (\alpha),\quad \| F\|_{B_{\delta (\alpha)}}<\infty \}, \] where \[ \| F\|_{B_{\Delta (\alpha)}}=\{\iint_{\Delta (\alpha)}| F(x+iy)|^ 2dxdy\}^{1/2}. \] Then, obtain Theorem 1. Let \(0<\alpha <\pi /2\). If \(F\in B_{\Delta (\alpha)}\), then \[ (1)\quad \iint_{\Delta (\alpha)}| F|^ 2dxdy=\sin (2\alpha)\sum^{\infty}_{j=0}\frac{(2 \sin \alpha)^{2j}}{(2j+1)!}\int^{\infty}_{0}x^{2j+1}| \partial^ jf(x)|^ 2dx, \] where f stands for the trace of F on the positive real axis. Conversely, if f is a smooth function on the positive real axis for which the right hand side of (1) is finite, then f has an analytic continuation \(F\in B_{\Delta (\alpha)}\) and (1) holds. Consider a counterpart of Theorem 1 for the Szegö space \(S_{\Delta (\alpha)}\) which is normed by the square root of \(\int_{\partial \Delta (\alpha)}| F(z)|^ 2| dz|\) with F(z) being the nontangential boundary values of F on \(\partial \Delta (\alpha).\) Then, we have Theorem 2. Let \(0<\alpha <\pi /2\). If \(F\in S_{\Delta (\alpha)}\), then \[ (2)\quad \int_{\partial \Delta (\alpha)}| F(z)|^ 2| dz| =2 \cos \alpha \sum^{\infty}_{j=0}\frac{(2 \sin \alpha)^{2j}}{(2j)!}\int^{\infty}_{0}x^{2j}| \partial^ jf(x)|^ 2dx, \] where f stands for the trace of F on the positive real axis. Conversely, if f is a smooth function on the positive real axis for which the right hand side of (2) is finite, then f has an analytic continuation \(F\in S_{\Delta (\alpha)}\) and (2) holds.