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.
Categories: math.AG