A few remarks about linear operators and disconnected open sets in the plane

Let E be a nonempty compact set with empty interior in the complex plane, and let U ⊆ C be a bounded open set such that U ∩ E = ∅ and ∂U = E. For instance, U might be the union of all of the bounded components of C\E. We are especially interested here in the situation where E is connected and U has infinitely many connected components, and moreover where each neighborhood of each element of E contains infinitely many components of U . Of course, U can have only countably many components, since the plane is separable. Sierpinski gaskets and carpets are basic examples of such fractal sets E. One can choose different orientations for the different components of U . This can be represented by a locally constant function on U with values ±1, where +1 corresponds to the standard orientation on C. Alternatively, a locally constant function on C\E with values 0 and ±1 can be used to specify which complementary components are included in U as well as their orientations. Bergman and Hardy spaces on U can then be defined using holomorphic or conjugate-holomorphic functions on the components of U , depending on the choice of orientation of the component. The topological activity in E indicated by its complementary components can also be described in terms of homotopy classes of continuous mappings from E into C\{0}. This is reflected in Fredholm indices of associated Toeplitz operators too. It is natural to allow different orientations on different components of U , in order to get different combinations of indices, which may involve many variations. At the same time, the standard orientation on all of U has some special features. Perhaps the first point is that there are a lot of holomorphic functions in the usual sense on neighborhoods of U , which are in particular very regular functions on U that are holomorphic on U . Conversely, removable singularity results imply that a sufficiently well-behaved function on U which is holomorphic on U is holomorphic on the interior of U . For instance, this holds for continuouslydifferentiable functions on U because E has empty interior, and for Lipschitz functions of order 1 on U when E has Lebesgue measure 0. Note that the differential of a continuously-differentiable function f on U which is holomorphic on U1 ⊆ U and conjugate-holomorphic on U2 ⊆ U vanishes on U1 ∩ U2. Since E is connected, U1 ∩ U2 6= ∅ when U1, U2 6= ∅ and U1 ∪ U2 = U .