Hardy spaces on Riemann surfaces under ramified coverings

We extend the theory of indefinite Hardy spaces on finite bordered Riemann surfaces to the setting of ramified analytic coverings. Given a finite $n$-sheeted ramified covering $F\text{\hspace{0.17em}⁣}:S_1\to S_2$ of finite bordered Riemann surfaces satisfying a spin-compatibility hypothesis, we construct (i) the direct image of a unitary flat vector bundle $\text{VxX}1\otimes \text{Del}1$ on the double $X_1$ under $F$, taking full account of the ramification divisor $R_F$ and establishing the extension across the branch locus via a careful local analysis; (ii) a canonical matrix function $G_2$ encoding the parahermitian structure on $X_2$, together with the induced representation ${\chi }_2$ of $\text{piX}{X}_2{p}_0$; (iii) an explicit isometric isomorphism ${\phi }_F\text{\hspace{0.17em}⁣}:H^{2,J_1(p)}(S_1,\text{VxS}1\otimes \text{Del}1)\stackrel{\sim }{\to }{H}^{2,J_2(p)}(S_2,\text{VxS}2\otimes \text{Del}2)$ between the associated Hardy-Kreĭn spaces, provided that $h^0(X_1,\text{VxX}1\otimes \text{Del}1)=0$ and that the branch locus is disjoint from $\mathrm{\partial }{S}_2$. We then develop the resulting operator theory in terms of vessels and Bezoutian operators. To each object in the category $\mathcal{R}\mathcal{H}$ of finite bordered surfaces with unitary flat bundles we attach a triangular vessel whose input and output spaces are the Hardy-Kreĭn spaces on the two surfaces; the Bezoutian of the vessel is expressed as a finite-rank operator on ${\mathcal{H}}_2$ whose kernel is built from bounded holomorphic point-evaluation functionals in ${\mathcal{H}}_2$ evaluated at the interior ramification images $F(r_{\nu })\in S_2$, consistently with the boundary-transversality hypothesis $\mathrm{\partial }{S}_2\cap B_F=\left\{\mathrm{\varnothing }\right\}$. We prove that the assignment $(S,\text{Vx},J)\mapsto H^{2,J(p)}(S,\text{Vx}\otimes \mathrm{\Delta })$ extends to a covariant functor from $\mathcal{R}\mathcal{H}$ (with ramified morphisms) to the category of Kreĭn spaces.