Hardy Spaces on Compact Riemann Surfaces with Boundary

We consider the holomorphic unramified mapping of two arbitrary finite bordered Riemann surfaces. Extending the map to the doubles $X_1$ and $X_2$ of Riemann surfaces we define the vector bundle on the second double as a direct image of the vector bundle on first double. %% We choose line bundles of half-order differentials $\Delta_1$ and $\Delta_2$ so that the vector bundle $V^{X_2}_{\chi_\ad} \otimes \Delta_2$ on $X_2$ would be the direct image of the vector bundle $V^{X_1}_{\chi_\ao} \otimes \Delta_1$. We then show that the Hardy spaces $H_{2, J_1(p)} (S_1,V_{\chi_\ao} \otimes \Delta_1)$ and $H_{2,J_2(p)} (S_2,V_{\chi_\ad} \otimes \Delta_2)$ are isometrically isomorphic. Proving that we construct an explicit isometric isomorphism and a matrix representation $\chi_\ad$ of the fundamental group $\piod(X_2, p_0)$ given a matrix representation $\chi_\ao$ of the fundamental group $\piod(X_1, p'_0)$. %% On the basis of the results of \cite{vin} and Theorem \ref{theorem_1} proven in the present work we then conjecture that there exists a covariant functor from the category ${\cal RH}$ of finite bordered surfaces with vector bundle and signature matrices to the category of Kre\u{\i}n spaces and isomorphisms which are ramified covering of Riemann surfaces.