The talk will try survey the recent trends in the theory of operators between Banach Spaces. In particular, we will show that there exist Banach spaces having essentially only compact operators between them, which are known to be very “small” (meaning that the range of such operators is a compact subset of the unit ball). This leads to a related question: Are there Banach Spaces with yet smaller family of operators? e.g. nuclear operators? We will review this and similar problems.
The presentation is meant to be self-contained, meaning that all important concepts will be introduced and explained. No special expertise in Banach Space Theory is assumed.