Toeplitz and Hankel operators on Hardy spaces of complex domains

The major focus is on the Hardy spaces of the annulus {z : s < |z| < 1}, with the measure on the boundary being Lebesgue measure normalised such that each boundary has weight 1. There is also consideration of higher order annuli, the Bergmann spaces and slit domains. The focus was on considering analogues of classical problems in the disc in multiply connected regions.
Firstly, a few factorisation results are established that will assist in later chapters. The Douglas-Rudin type factorisation is an analogue of factorisation in the disc, and the factorisation of H1 into H2 functions are analogues of factorisation in the disc, whereas the multiplicative factorisation is specific to multiply connected domains.
The Douglas-Rudin type factorisation is a classical result for the Hardy space of the disc, here it is shown for the domain {z : s < |z| < 1}. A previous factorisation for H1 into H2 functions exists in [4], an improved constant not depending on s is found here.
We proceed to investigate real-valued Toeplitz operators in the annulus, focusing on eigenvalues and eigenfunctions, including for higher order annuli, and amongst other results the general form of an eigenfunction is determined. A paper of Broschinski [10] details the same approach for the annulus {z : s < |z| < 1} as here, but does not consider higher genus settings. There exists work such as in [6] and [5] detailing an alternative approach to eigenvalues in a general setting, using theta-functions, and does not detail the eigenfunctions.
After this, kernels of a more general symbol are considered, compared to the disc, and Dyakanov’s theorem from the disc is extended for the annulus.
Hankel operators are also considered, in particular with regards to optimal symbols. Finally, analogues of results from previous chapters are considered in the Bergman space, and the Hardy space of a slit annulus.