Two classical problems on the shift operator in the Bloch space

In this talk, I intend to discuss two classical problems related to the shift operator acting on the classical Bloch space in the unit disc. The Bloch space has a rich history which originates back to the work of Bloch, Landau and Valiron in the 1920’s, on right inverses of holomorphic mappings. The space of Bloch functions is also very intimately related to the theory of conformal mappings. At the end of the 1980’s, Makarov established a deep connection between Bloch functions and dyadic martingales with uniformly bounded jumps, which further enhanced the development of probabilistic tools for the purpose of analyzing boundary behaviors of conformal maps. As the Bloch space in the setting of Bergman spaces is the analogue of BMOA in the classical Hardy spaces, many natural problems in operator and function theory, such as describing zero sets, interpolating sequences and invariant subspaces of various operators, may also be phrased therein. However, it turns out that the relationship between the Bergman spaces and the Bloch space deviates substantially from the corresponding relationship between Hardy spaces and BMOA. Our purpose is to specifically illustrate these points in the context of two seemingly different problems related to the shift operator in the Bloch spaces, namely the problem on shift invariant subspaces and the problem of simultaneous approximation in the Bloch space. We intend to answer a couple of questions left open in the beginning of the 1990’s by Anderson, Brown, Shields and others. Parts of this talk is based on joint recent work with Artur Nicolau.