Isometrical identities for the Bergman and the Szegö spaces on a sector

In the case of $\alpha=\pi/4$ we showed that $||F||_{B}^{z_{\Delta(a)}}$ is represented as a series of weighted square integrals of the derivatives of the trace of $F$ on the positive real axis ([2]). The proof included two different ingredients: an integral transform and a heat equation on the positive real axis. Both of them required rather deep and lengthy arguments which worked only in the case of $\alpha=\pi/4$ . Here we present a general result for $0<\alpha<\pi/2$ by a completely different proof with minimum prerequisite knowledge. We shall show THEOREM 1. Let $0<\alpha<\pi/2$ . If $F\in B_{\Delta(a)}$ , then