Exercise 8, § 3

Let U be a connected open set of $\mathbf{C}$. We denote by $\mathscr{H}(U)$ the space of holomorphic functions on U endowed with the topology of compact convergence (EVT III, p. 10, example d)).

a) Let $a\in \mathscr{H}(U)$. Multiplication by $a$ is a Fredholm endomorphism of $\mathscr{H}(U)$ if and only if the set of zeros of $a$ is finite. Calculate its index in that case.

b) Let $\mathscr{U}= (U_i)_{i\in I}$ be a covering of U by non-empty connected open sets. We put

We consider the vector spaces

$Z^1(\mathscr{U};\mathbf{C}) =\{(c_{i,j})\in \mathbf{C}^{I_2}|c_{ij}+c_{jk}+c_{ki}= 0$ for all $(i, j, k)\in I_3\}$

$B^1(\mathscr{U};\mathbf{C}) =\{(c_{i,j})\in \mathbf{C}^{I_2}|$ there exists $c'\in \mathbf{C}^I$ such that

$c_{ij}=c'_i-c'_j$ for all $(i, j)\in I_2\}$.

We have $B^1(\mathscr{U};\mathbf{C})\subset Z^1(\mathscr{U};\mathbf{C})$. We put

Let $\mathscr{V}= (V_{\alpha})_{\alpha\in A}$ be a covering of U by non-empty connected open sets, finer than $\mathscr{U}$, and let $\varrho : A\rightarrow I$ be a mapping such that $V_{\alpha}\subset U_{\varrho(\alpha)}$ for all $\alpha \in A$.

The mapping $(c_{ij})_{i,j}\mapsto (c_{\varrho(\alpha)\varrho(\beta)})_{\alpha ,\beta}$ induces a linear mapping, independent of the choice of $\varrho$, from $H^1(\mathscr{U};\mathbf{C})$ into $H^1(\mathscr{V};\mathbf{C})$.

We denote by $H^1(U;\mathbf{C})$ the inductive limit of the vector spaces $H^1(\mathscr{U};\mathbf{C})$ endowed with these linear mappings, when $\mathscr{U}$ runs through the set of coverings of U by non-empty connected open sets.

c) Let D be the endomorphism of $\mathscr{H}(U)$ defined by $Df=f'$. Prove that there exists an isomorphism of vector spaces from the cokernel of D onto $H^1(U;\mathbf{C})$. Deduce that D is a Fredholm endomorphism if and only if the space $H^1(U;\mathbf{C})$ is of finite dimension. Calculate then the index of D.