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 ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$ so that the vector bundle $V_{{\chi }_{\text{ad}}}^{X_2}\otimes {\mathrm{\Delta }}_2$ on $X_2$ would be the direct image of the vector bundle $V_{{\chi }_{\text{ao}}}^{X_1}\otimes {\mathrm{\Delta }}_1$. We then show that the Hardy spaces $H_{2,J_1(p)}(S_1,V_{{\chi }_{\text{ao}}}\otimes {\mathrm{\Delta }}_1)$ and $H_{2,J_2(p)}(S_2,V_{{\chi }_{\text{ad}}}\otimes {\mathrm{\Delta }}_2)$ are isometrically isomorphic. Proving that we construct an explicit isometric isomorphism and a matrix representation ${\chi }_{\text{ad}}$ of the fundamental group $\text{piod}(X_2,p_0)$ given a matrix representation ${\chi }_{\text{ao}}$ 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 $\mathcal{R}\mathcal{H}$ 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.