Medical, Pharma, Engineering, Science, Technology and Business

**Zuevsky A ^{*}**

School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland

- Corresponding Author:
- Zuevsky A

School of Mathematics, Statistics and Applied Mathematics

National University of Ireland, Galway, Ireland

**Tel:**3531439 2424

**E-mail:**[email protected]

**Received date** July 21, 2015; **Accepted date** August 02, 2015; **Published date** August 31, 2015

**Citation:** Zuevsky A (2015) Hardy Spaces on Compact Riemann Surfaces with Boundary. J Generalized Lie Theory Appl S1: 005. doi:10.4172/1736-4337.S1-005

**Copyright:** © 2015 Zuevsky A. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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We consider the holomorphic unramified mapping of two arbitrary finite bordered Riemann surfaces. Extending the map to the doubles X1 and X2 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 Δ1 and Δ2 so that the vector bundle 2 χ2 2 V X ⊗Δ on X2 would be the direct image of the vector bundle 1 1 χ ⊗Δ2 V X . We then show that the Hardy spaces 2, 1( ) ( 1, χ1 1) H J p S V ⊗Δ and 2, 2 ( ) ( 2, χ2 2) H J p S V ⊗Δ are isometrically isomorphic. Proving that we construct an explicit isometric isomorphism and a matrix representation χ2 of the fundamental group π1(X2, p0) given a matrix representation χ1 of the fundamental group π1(X1, p'0). On the basis of the results of Alpay et al. and Theorem 3.1 proven in the present work we then conjecture that there exists a covariant functor from the category ï��ï�� of finite bordered surfaces with vector bundle and signature matrices to the category of KreÄn spaces and isomorphisms which are ramified covering of Riemann surfaces.

Half-order differentials; Hardy spaces; Riemann surfaces

It is well known how to study Hardy spaces defined on a finite bordered Riemann surface [1-5]. For domains with more than one boundary component it is natural to introduce, besides the usual positive definite inner product on H2, indefinite inner products. Those products may be introduced by picking up different signature matrices when integrating over different components of the boundary of the Riemann surface. In the paper Alpay et al. [1] a necessary and sufficient condition for such an indefinite inner product to be non-degenerate was obtained. It was shown that when this condition is satisfied one actually gets a Kreĭn space. The result was obtained by using a covering map of the surface to the unit disk to construct an isomorphism to a Hardy-Kreĭn space over the unit disk. Furthermore, each holomorphic mapping of the finite bordered Riemann surface onto the unit disk (which maps boundary to boundary) determines an explicit isometric isomorphism between this space and a usual vector-valued Hardy space on the unit disk with an indefinite inner product defined by an appropriate hermitian matrix. The mapping to the unit disk in Alpay et al. [1] serves as a tool to study of the Hardy-Kreĭn space over the finite bordered Riemann surface which in turn has motivation from the point of view of the study of commuting tuples of non-self adjoint operators. As it is usual when studying Hardy spaces on a multiply connected domain, the elements of the space are sections of a vector bundle rather than functions. The main point of the paper by Alpay et al. [1] was to construct an appropriate extension of this bundle to the double of the finite bordered Riemann surface and to use Cauchy kernels for certain vector bundles on a compact Riemann surface. Hardy spaces on a finite bordered Riemann surface, including indefinite Hardy spaces, are important in the model theory for commuting non-self adjoint operators [6].

Half-order differentials play a very important role in the vertex opera-tor algebra approach to construction of partition and n-point functions for conformal field theories defined on Riemann surfaces [7-9]. In particular, the Szegö kernel [10] turned out to be key object in construction of correlation functions in free fermion conformal field theories/chiral algebras on a genus two Riemann surface sewed from two genus one Riemann surfaces [11].

In this work we replace a holomorphic mapping of a finite bordered Riemann surface onto the unit disk by a holomorphic mapping of two arbitrary finite bordered Riemann surfaces S1 and S_{2}, which we assume however to be unramified. In the spirit of Alpay et al. [1,12] one can introduce the extension of the vector bundles on a finite bordered Riemann surfaces to the respective doubles. Extending the map F to the doubles X_{1} and X_{2} of Riemann surfaces S1 and S_{2} we define the vector bundle on X_{2} as a direct image of the vector bundle over X_{1}. We choose line bundles of half-order differentials (i.e., square roots of the canonical bundles , i = 1, 2) Δ_{1} and Δ_{2} so that the vector bundle on X_{2} would be the direct image of the vector bundle . We then show that the Hardy spaces bundle and are isometrically isomorphic. Proving that we construct a.) an explicit isometric isomorphism; b.) a matrix representation χ_{2} of the fundamental group π_{1}(X_{2}, p_{0}) given a matrix representation χ_{1} of the fundamental group π_{1}(X_{1}, p′_{0}). Using results of Alpay et al. [1] and Theorem 3.1 proven in the present work we then conjecture that there exists a covariant functor from the category **RH** of finite bordered surfaces with vector bundle and signature matrices to the category of Kreĭn spaces and isomorphisms which are ramified covering of Riemann surfaces.

The isomorphism established in this work has also an operator theoretical interpretation, namely, a (ramified) covering F allows us to construct a pair of commuting non-self adjoint operators with the model space on S_{2} given a pair of commuting non-self adjoint operators with the model space on S1. More generally, one might expect also possible connections with vessels construction and Bezoutians [13].

**Preliminaries**

As we mentioned in Introduction indefinite *Hardy spaces* [14-16] on a finite bordered Riemann surface were considered in Alpay et al. [1].

**Definition.** Let J be an m x m unitary self-adjoint matrix. Such a matrix is usually called a *signature matrix*. (In fact one may take J to be any non-singular self-adjoint matrix). The Hardy space on the unit disk D endowed with the indefinite inner product

is a Kreĭn space denoted , J .

This space plays an important role in interpolation theory [17] and in model theory [18]. For the general theory of Kreĭn spaces[19-21].

Suppose now that we have an open Riemann surface S such that S ∪ ∂S is a finite bordered Riemann surface (i.e., a compact Riemann surface with boundary), with the boundary ∂S consisting of k ≥ 1 components . We consider analytic sections of a rank m flat unitary vector bundle Vχ on S corresponding to a homomorphism χ from the fundamental group π_{1}(S, p_{0}) into the group U(m) of m × m unitary matrices. An analytic section f of V_{χ} over S is an analytic C^{m} valued function on the universal covering of S satisfying

for all and all deck transformations T of over S, which we identify with elements of the fundamental group π_{1}(S, p); f can be thought of as a multiplicative multivalued function on S. We consider also multiplicative half-order differentials [1], i.e., sections of a vector bundle of the form ,where Vχ is a flat unitary vector bundle on S as above and Δ is a square root of the canonical bundle on

**Definition.** The Hardy space on a Riemann surface S is the set of sections of a vector bundle analytic in S satisfying

(1)

for some ϵ > 0. In (1) (r) denotes smooth simple closed curves in S approximating , i = 0, ..., k - 1 ([1]): if z_{i} is a boundary uniformizer near the boundary component then (r) is given by |z_{i}(p)| = r.

Note that since is a section of , the expression is a section of |K_{S}|, where |K_{S}| is the line bundle with transition functions the absolute values of the transition functions of KS ; sections of |KS| can be represented locally as η(t)|dt(p)| where t(p) is a local parameter. Therefore one can integrate over curves in S and (1) makes sense.

The space is a Hilbert space with the inner product

For a relation between and Hardy spaces of functions on S with respect to a harmonic measure on ∂S, see Alpay et al. [1].

**Definition.** Denote by the analogue of the Kreĭn space for S which is the Hardy space endowed with the indefinite inner product

(2)

where is the non-tangential boundary values of which exists again almost everywhere on ∂S (Alpay et al.[1]) and is a locally constant matrix function on whose values are m x m signature matrices, satisfying

(3)

for all and all . The expression in (2) means where is over . It is a welldefined section of |KS| because of the transformation property of . There exists certain freedom in the choice of for the given Vχ . Indeed, choose points . Let C_{i} be a path on S linking p_{0} to p_{i}. Set (see Appendix). Then the (homotopy class of) C_{i} determines a component of lying over , and the constant value Ji of on can be an arbitrary m x m signature matrix satisfying

Any other component of lying over can be obtained from by some deck transformation R. The value of on this component is . For the case of the line bundles (i.e., m = 1), the choice of amounts to an arbitrary choice of a sign ±1 for each . We will often assume the choice of components has been made and denote by The space is a natural example of an indefinite inner product space. It is related to the model theory of pairs of commuting non-selfadjoint operators and interpolation theory on multiply connected domains.

In the paper Alpay et al. [1] an appropriate extension of Vχ on S to the double X of the Riemann surface S was constructed. Given a flat unitary vector bundle on a finite bordered Riemann surface, together with a collection of signature matrices, it can be uniquely extended to a flat unitary vector bundle on the double satisfying certain symmetry properties. Let us recall that construction.

Due to the identification of the boundaries the complex structures on two copies of S constituting X are mirror images of each other, i.e., there exists an *anti-holomorphic involution* τ : X → X that maps to . Thus X is a compact real Riemann surface, or equivalently Riemann surface of a real algebraic curve. The genus g of the double of X of S is g = 2s + k − 1, where s is the genus of S. The set X_{f} of fixed points of τ (real points of X) coincides with the boundary ∂S of S. Furthermore X is a real Riemann surface of dividing type: the complement X \ X_{f} consists of two connected components X+ = S and X− = S′ interchanged by τ. The converse is also true: any real Riemann surface of dividing type is the double of a finite bordered Riemann surface. The anti-holomorphic involution τ acts both on the fundamental group π_{1}(X, p_{0}) and on the universal covering of X (recall that the fundamental group π1(X, p_{0}) is isomorphic to the group of deck transformations Deck ( /X). It also acts naturally on complex holomorphic vector bundles on X: the transition functions for the vector bundle V τ complex conjugates of the transition functions for V at the point conjugate under τ.

Consider a vector bundle H on X of rank *m* with deg H = m(g-1) satisfying the condition h^{0}(H) = 0. Such a vector bundle is necessarily of the form where Vχ is a rank m flat vector bundle on X and Δ is a square root of KX [22]. These vector bundles are closely related to determinantal representations of algebraic curves and play an important role in the theory of commuting non-self adjoint operators and related theory of 2D systems [22-26].

Let H be such that there exists a non-degenerate bilinear pairing H × H^{τ} → K_{X} which is parahermitian. The parahermitian property means that

for all local holomorphic sections and of H near p and pτ respectively. We assume also that the line bundle Δ has been chosen so that and that the transition functions of Δ are symmetric with respect to τ [27]; Then

we obtain a parahermitian non-degenerate bilinear pairing or more explicitly an everywhere nonsingular holomorphic m x m matrix–valued function G on the universal covering with the property

(4)

satisfying the relation

(5)

where T ∈ π_{1}(X, p_{0}). The pairing H × H^{τ} → K_{X} is then given explicitly by

Now let us introduce the (in general) indefinite inner product

(6)

where and are measurable sections of H over X_{f} . Here and in similar expressions, the integral is computed on X and the integrand does not depend on the choice of above p ∈ X Since in (6) lies over a point of Xf there exists such that . Therefore (6) can be rewritten as

(7)

Where

Note that and

(8)

for all over X_{f} and all R ∈ π_{1}(X, p_{0}). Thus the vector bundle on X defines an indefinite inner product on the sections of its restriction to X_{f} = ∂S.

For lying over a point of , we have for i = 0 and for i = 1, . . . , k − 1 where are part of the generators of the fundamental group π_{1}(X, p_{0}) of X (refer Appendix for the relation between generators of the fundamental groups of a Riemann surface S and the corresponding double), and R depends only on the component of the inverse image of in that belongs to. Restricting in (7) to belong to a specific component we may write

for i = 0 and

for i = 1, . . . , k − 1 (refer Appendix). (The specific component depends on the choice of the generators B_{i}, i.e., on the homotopy classes of the paths C_{i}.)

It follows from the conditions deg H = m(g −1) and h^{0}(H) = 0 that H is a semi-stable vector bundle. By a theorem of Narasimhan and Seshadri [28] H is a direct sum of stable bundles if and only if the flat vector bundle V_{χ} (in ) can be taken to be unitary flat. Since G is an isomorphism from Vχ to the dual of it follows in this case that G is constant and unitary. Since it is also selfadjoint, it is a constant signature matrix. Thus for analytic sections and of on S that belong to we can rewrite the inner product (7) as

(9)

where

J_{0} = G, J_{i} = χ(B_{i})*G,

for i = 1, . . . , k−1 and p is restricted to belong to a specific component of the inverse image of χ_{i} in as explained above. We then obtain for the vector bundle H on X the inner product (2) on the Hardy space . Conversely, every unitary flat vector bundle on S with signature matrices J_{0}, . . . , J_{k−1} can be obtained from a vector bundle H on X as above. Let J_{i}, i = 0,..., k − 1 be self adjoint matrices and let be a homomorphism satisfying

Then by Proposition 2.1 from Alpay et al. [1] there exists a unique extension (still denoted by χ) of χ to a homomorphism from π_{1}(X, p_{0}) into GL(m, ) satisfying

where G = J_{0} (refer Appendix).

If the original flat vector bundle Vχ on S is unitary flat and all the matrices Ji are unitary then the extended vector bundle is also unitary flat as it is follows form the proof of Proposition 2.1 of Alpay et al. [1]. The extension need not satisfy ; i.e., the unitary case, this condition will be satisfied ”generically” since flat unitary vector bundles Vχ on X with form a divisor in the moduli space of flat unitary vector bundles (the generalized theta divisor [10,29]). It was proven in Proposition 2.2 from Alpay et al. [1] that if the indefinite inner product space is nondegenerate, then

(10)

It follows that the condition (10) is satisfied automatically in the positive definite case (i.e., when J_{i} > 0 for i = 0, . . . , k − 1); for line bundles, this has been obtained in Fay et al. [27,30].

Summing up, we see that the above extension procedure establishes a one-to-one correspondence between unitary flat vector bundles on S together with a choice of signature matrices satisfying (3) and unitary flat vector bundle on X satisfying the symmetry condition (4), (5). Given a unitary flat vector bundle on S, the various choices of extension to the double X correspond to the various choices of signature matrices. We shall occasionally denote the corresponding unitary flat vector bundles on S and X by and respectively.

Under the condition (i.e., that has no global holomorphic sections), it turns out that on X admits a certain kernel function (which is called the Cauchy kernel) which is an analogue of for the trivial bundle on the complex plane. The Cauchy kernel is the reproducing kernel for . In the case of line bundles the Cauchy kernel can be given explicitly in terms of theta functions [27,31]. In Alpay et al. [1] the Cauchy kernel was used to construct for any given holomorphic mapping z : S → D an explicit isometric isomorphism between and for appropriate M and J. In particular this implies that is indeed non-degenerate (under the condition ) and actually a Kreĭn space.

Suppose that we have two finite bordered Riemann surfaces S1 and S_{2}. Let F: S1 → S_{2} be an analytic mapping continuous up to the boundary.

Equivalently we may take F to be a complex analytic mapping F: X_{1} → X_{2} between the doubles of S1 and S_{2 }equivariant with respect to the action of the anti-holomorphic involutions, i.e., such that the diagram

is commutative. Notice that F: S1 → S_{2} is unramified if and only if F: X_{1} → X_{2} is unramified.

We identify as usual a complex holomorphic vector bundle on a complex manifold with a locally free sheaf of its analytic sections. It is easily seen that if V^{X} is a complex holomorphic vector bundle of rank m on a complex manifold X and F is a n-sheeted unramified covering, then the direct image V^{Y} = F*V^{X} is a complex holomorphic vector bundle of rank nm on Y and the fiber of V^{Y} at a given point of Y is the direct sum of the fibers of V^{X} at the preimages of this point on X.

The main statement of this work is the following

**Theorem 3.1*** Let F: S1 → S _{2} be a map of finite bordered Riemann surfaces which is a finite n-sheeted unramified covering (F ; S1, S_{2}), and let be signature matrices for a unitary flat vector bundle on S1 of rank m. Consider the corresponding extension of to the double X_{1} of S1 satisfying the symmetry condition*

for all R ∈ π_{1}(X_{1}, p′_{0}) and all . Choose the bundles Δ_{1} and Δ_{2} of half-order differentials on X_{1} and X_{2} respectively, such that

a.) the bundles Δ_{i}, i = 1, 2 are invariant with respect to the corresponding anti-holomorphic involutions, i.e., and the transition functions of Δ_{1} and Δ_{2} are symmetric with respect to τ_{1} and τ_{2};

b.) the pull–back of Δ_{2} is equal to Δ_{1}, i.e., Δ_{1} = F*Δ_{2}.

Then

1) the direct image is a unitary flat holomorpnic vector bundle of rank nm satisfying the symmetry condition

for all T ∈ π_{1}(X_{2}, p_{0}) and all , appropriate matrix function , and representation χ2 of π_{1}(X_{2}, p_{0}); furthermore *; *

2) there exists a canonical isometric isomorphism

between Hardy spaces on S1 and S_{2}.

Now some remarks are in order. By definition the anti-holomorphic involutions τ_{1} and τ_{2} are related by

F ○ τ1 = τ2 ○ F,

and therefore if we have a line bundle L_{2} on X_{2} then its pull-back satisfies

We fix Δ_{2} such that and . Then it follows that . We choose Δ_{2} such that its transition functions are symmetrical. Then since F is equivariant with respect to the antiholomorphic involution then transition functions of Δ_{1} are also symmetrical.

The isomorphism of the spaces and implies that they are degenerate or nondegenerate simultaneously, i.e., which is obvious from the definition of the direct image vector bundle.

We assume that the map F: S1 → S_{2} is a n-sheeted unramified covering (F; S1, S_{2}) of the Riemann surface S_{2} by S1. On the other hand, a result of Alpay et al. [1] mentioned in Introduction is a construction of an isometric isomorphism between Hardy spaces when S_{2} = D but F is (usually) ramified (assuming is not degenerate, i.e., . The next natural step would be to consider the case when S_{2} is an arbitrary finite bordered Riemann surface and F is a ramified covering. That will be a point of some further publication.

We have introduced the vector bundle on the double X_{2} as the direct image of the vector bundle on X_{1} defining the vector bundle on S_{2} and the signature matrices . On the other hand one can define the vector bundle to be the direct image of the vector bundle with signature matrices defined naturally in terms of (as direct sums). Though the main claim of Theorem 3.1 is formulated for finite bordered Riemann surfaces it seems to us that the consideration of the structures involved in its proof is more natural (in the sense of the theory of compact Riemann surfaces) on the doubles. Furthermore, this approach allows us to construct a matrix representation of the fundamental group π_{1}(X_{2}, p_{0}) given a representation of π_{1}(X_{1}, p′_{0}), and the matrix function . We will prove that signature matrices calculated with the help of the representation χ2 do coincide with the signature matrices constructed directly from the signature matrices . This shows the equivalence of those two approaches. From the use of Cauchy kernels in Alpay et al. [1] it seems however that in the ramified case the approach via the doubles is the only one possible.

Speaking in more abstract terms we deal in Theorem 3.1 with a category which we will denote by . Objects of are finite bordered Riemann surfaces S together with a unitary flat vector bundle Vχ and signature matrices (or equivalently, compact real Riemann surfaces X of dividing type with a vector bundle on X and a matrix functionon X and a matrix function satisfying (4) and (5)) such that the space is non degenerate, i.e., . A morphism between the objects and of is an analytic map F : X_{1}→X_{2} of Riemann surfaces which is equivariant with respect to anti-holomorphic involutions τ_{1} and τ_{2}, such that (and is correspondingly induced by ).

We conjecture that there exists a covariant functor from the above men-tioned category to the category of Krein spaces and isomorphisms, asso-ciating to the Hardy space .

Theorem 3.1 proves the conjecture for a subcategory of whose morphisms are unramified coverings. The isometric isomorphism established in Alpay et al. [1] proves another special case of the conjecture namely for a subcategory whose morphisms are restricted to have the unit disk D as a range. Somewhat related considerations of categories of functional spaces on Riemann surfaces are contained in Alling et al. [32].

In this section we give an explicit construction of a holomorphic section of the bundle on X_{2} in terms of a holomorphic section of on X_{1}. When (F; S1, S_{2}) is an unramified covering the doubles X_{1} and X_{2} possess the common universal covering , i.e., one has a diagram

where π_{1} and π_{2} are the covering maps from to X_{1} and X_{2} respectively.

Let U′ ⊂ X_{1} be an open set in X_{1}. Suppose f^{1} is an analytic section of the holomorphic vector bundle over U′, i.e., an analytic valued function on satisfying the relation

(11)

Similarly, a section of the vector bundle over U′ satisfies

for all , where t_{1} is a local parameter on X_{1} lifted to . The fundamental group , is a subgroup of π_{1}(X_{2}, p_{0}) of index n (here p_{0}′ is a preimage of p_{0} ∈ X_{2}). Enumerate fixed representatives g_{i}, i = 1,..., n of the left cosets {π_{1}(X_{1}, p_{0}′ )g_{i}} of the group π_{1}(X_{2}, p_{0}) with respect to its subgroup π_{1}(X_{1}, p′). We define a sheaf on X_{2} whose sections over an open set U⊂ X_{2} are analytic valued functions on of the vector form

i = 1,…, n, where f1 is a section of the bundle over F^{-1}(U), i.e., is an analytic -valued function on satisfying (11). It easy to see from the definition that this sheaf on X_{2} is isomorphic to the direct image sheaf of a sheaf on X_{2} of analytic sections of , i.e., (12) defines the sheaf of analytic sections of F*.

Now let p ∈ X_{2} and p′_{1}, ..., p′_{n} ∈ X_{1} be preimages of p. Let t_{2} and t_{1,i} be local parameters near p and p′_{i}, i = 1, ..., n lifted to the common universal covering . Denote by φ_{i} the composition . Then a section of a vector bundle is given by

(13)

where is a section of the vector bundle and t_{1,i}, i = 1, ..., n are local parameters in the vicinity of . Since we have chosen the bundles Δ_{1} and Δ_{2} of half-order differentials in (13), the ambiguity in the sign of the square roots of in (13) is global and since we have assumed that Δ_{1} = F *Δ_{2}, the expression (13) does not depend on the choice of local parameters.

In this section we give an explicit formula for a unitary representation χ2 of 2 π_{1}(X_{2}, p_{0}) such that . It follows from the previous section that we have to define χ2 so that

(14)

for every f^{2} given by (12). Let g ∈ π_{1}(X_{2}, p_{0}). Fix a preimage p′_{0} ∈ X_{1} of p_{0}. The element g belongs to a coset of the fundamental group π_{1}(X_{2}, p_{0}) with respect to its subgroup π_{1}(X_{1}, p′_{0}). Then there exist elements h ∈ π_{1}(X_{1}, p′_{0}) and such that

i.e., g defines a permutation σ_{g} of the preimages of p_{0}. We take this as a definition of σ_{g}. We define the matrix representation χ2 as follows:

(15)

It is immediate that (14) is verified. Taking into account the unitarity of χ1(g) , it can be seen from (15) that the matrices defining the representation of π_{1}(X_{2}, p_{0}) are unitary, i.e.,

Now we check that (15) provides a representation π_{1}(X_{2}, p_{0}), i.e.,

for all Proving this we used the fact that χ1 is homomorphism and σg is an anti-homomorphism, i.e.,

which can be easily verified. In general, the matrix χ2 is given by the formula

where g belongs to i-th coset.

Suppose that H^{1} = is such that there exists a non-degenerate bilinear pairing which is parahermitian, i.e.,

We assume that the line bundle Δ_{1} is such that and the transition functions of Δ are symmetric with respect to τ1. Then we have a parahermitian non-degenerate bilinear pairing and the matrix function satisfying (4) and (5). One can define a bilinear non-degenerate pairing where introducing an everywhere nonsingular holomorphic mn × mn matrix-valued function G_{2} on the universal covering of X_{1} and X_{2}. The matrix should have the property

(16)

and satisfy the symmetry condition, T ∈ π_{1}(X_{2}, p_{0}) ,

(17)

Then the pairing is given by

Taking into account the explicit form (15) of χ2(g) one can check that the following expression for does satisfies (17)

(18)

where ν(k) is defined as follows. Consider the action of τ on an element g ∈ π_{1}(X_{2}, p_{0}). By definition we have g^{τ} = τgτ^{−1}. For any g^{k} that belongs to k-th coset of π_{1}(X_{2}, p) with respect to π_{1}(X_{1}, p′_{0}) there exist h_{k} ∈ π_{1}(X_{1}, p′_{0}) and g_{ν(k)} ∈ π_{1}(X_{2}, p) such that

(19)

We define ν(k) by (19). One can check directly that (18) does satisfy conditions (16) and (17).

We saw in Introduction how to define an indefinite inner product (2), (9) on the Hardy space using signature matrices . Suppose that we have such an inner product on

Then we define an indefinite inner product on

(20)

By the same reasons as in Introduction we can rewrite (20) as

(21)

Where

and introduce the matrices

(22)

where (refer Appendix). As in Alpay et al. [1] the extension of the bundle on the Riemann surface S_{2} to the double X_{2} depends on the choice of the signature matrices J_{2,0, ...}, J_{2,k−1} given by (22) and which satisfies the symmetry condition (8). On the other hand, one can define the signature matrix using the signature matrix . One should have

(23)

for all T ∈ π_{1}(X_{2}, p) and

(24)

for all over p ∈ X_{2,f}. The matrix in the form

(25)

satisfies (23) and (24). Then we check the commutativity of the diagram

Where ext, Ji means the extension of the vector bundle on S_{i} to the double X_{i} . I.e., we will show that the matrix defined by (22) coincides with (25). It easy to check that

for all R ∈ π_{1}(X_{2}, p), p ∈ X_{2,f} and such that where lies over p. Using that we arrive at

We have constructed explicitly a section of the bundle in terms of a section of the bundle . Now we will prove that the map is an isometric isomorphism of the space on the space . First let us show that if and only if .

Suppose is a section of the bundle and . That means that , i.e., is an analytic in X_{1}

And

for some ϵ. Here are smooth simple curves in X_{1} approximating the i-th boundary of the X_{1}. The space is the space endowed with the indefinite inner product (2)

Let be a boundary component of X_{2} and j = 1, ..., ni be corresponding preimages on X_{1}. The boundary uniformizer z_{2} near the boundary component is such that z_{2}p0′ = z2°Fp0′ . Then the approximating curves are mapped to the approximating curves . Due to the construction given by the formula (13) we see that is an analytic and

(26)

The summation in (26) with upper limits ni is performed over the components that are preimages of . Thus we infer that belongs to the space In the previous section we have introduced an indefinite inner product in the space Thus we see that a section of xconstructed by the formula (13) belongs to the space

Finally, it remains to show that the inner product (21) is isometric, i.e., that

(27)

where and are sections of the vector bundles and respectively. Indeed, consider the inner product of two sections of the bundle

By the same argument that were used in the formulae (26) the last integral is equal to

where we use the invariance of sections of Δ_{1} with respect to deck transformations and the symmetry of the their transition functions. Hence we see that (27) holds. That completes the proof of the isometricity.

Let us describe explicitly [1] the action of τ on the generators of π_{1}(X, p_{0}). Choose points and let C_{i} be a path on S linking p_{0} to p_{i}. Then π_{1}(S, p_{0}) is generated by

(28)

where for j = 1,…,k - 1 represent a canonical homology basis on S with the intersection matrix

. The generators of π_{1}(S, p_{0}) satisfy a single relation

Now consider the fundamental group π_{1}(X, p_{0}). It is generated by

The generators A_{j} , A′_{i}, B_{i}′ are the same as in (28)

for j = 1,..., k − 1 and

The generators of π_{1}(X, p_{0}) satisfy a single relation by Natanzon et al. [33-41]

Note that

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