Hankel operators on vector-valued Bergman spaces

Abstract

The main goal of this work is to study vector-valued Bergman spaces and to obtain the weak factorization of these spaces. In order to do that we need to study small Hankel operators with operator-valued holomorphic symbols. We also study the big Hankel operator acting on vector-valued Bergman spaces.

In Chapter 1 we collect all the previous results and notations needed to follow the rest of the manuscript. More concretely, some of the topics covered in this chapter are the Bochner integral, the integral for vector-valued functions appearing first in Bochner; the Bergman metric, results of the metric used in Bn; harmonic and subharmonic function; basic notions of differentiation, where the differential operators R(a, t) are presented which is important in the next chapters and in the final section we recall some topics on Banach spaces, as the Rademacher type and cotype of a Banach space and some other related results.

Having all that in mind, in Chapter 2, the vector-valued Bergman spaces are presented. The vector-valued Bloch type spaces play a similar role and therefore we dedícate one full chapter to these spaces. Chapter 3 is devoted to present and characterize the vector-valued Bloch type spaces. Since we mention Hankel operators, in Chapter 4 we prove the characterization of the boundedness of the small Hankel operator with analytic operator-valued symbols between vector-valued Bergman spaces (of different type). We explain what this means in the following. Another very important consequence of the boundedness of the small Hankel operator between vector-valued Bergman spaces is shown in Chapter 5. We establish the weak factorization of the vector-valued Bergman spaces. Factorization of analytic functions is a very big topic and many people worked on it during many years and it is known to have many applications.

Therefore, in Chapter 6 we fully characterize the boundedness of the big Hankel operator on vector-valued Bergman spaces in terms of its operator-valued holomorphic symbol for all cases of p > 1 and q > 1, and so we solve and generalize the previous problem.

Finally, in Chapter 7 we discuss some open problems we have not been able to solve, as well as some other interesting problems in the same line as this work in order to look on the future.