We classify self-adjoint first-order differential operators on weighted Bergman spaces the unit disc and answer questions related to uncertainty principles for such operators. Our main tools are discrete series representations of $$\textrm{SU}(1,1)$$ . This approach has promise generalize other bounded symmetric domains.

These are the notes of the lecture course given at LTCC in 2015. The aim of the course is to consider the following three classes of operators: Toeplitz and Hankel operators on the Hardy space on the unit circle and Toeplitz operators on the Bergman space on the unit disk. For each of these three classes of operators, we consider the following questions: boundedness and estimates or explicit ex...

We study reducing subspaces for an analytic multiplication operator Mzn on the Bergman space L 2 a(Ar) of the annulus Ar, and we prove that Mzn has exactly 2 reducing subspaces. Furthermore, in contrast to what happens for the disk, the same is true for the Hardy space on the annulus. Finally, we extend the results to certain bilateral weighted shifts, and interpret the results in the context o...