Operator theory on Banach spaces

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In 1980, Bourgain and Delbaen devised a revolutionary new way of constructing script L spaces. At the time, it attracted much attention, but over the following decades perhaps it faded somewhat into the background until about 15 years ago, when Argyros and Haydon successfully managed to fuse it with the construction of hereditarily indecomposable Banach spaces, originally due to Gowers and Maurey, to produce a Banach space on which every bounded operator is the sum of a compact operator and a scalar multiple of the identity. This solved the so-called "scalar-plus-compact'' problem, which was the pre-eminent open problem in Banach space theory at the time. The time is therefore ripe to investigate bounded operators on Banach spaces constructed via the Bourgain-Delbaen method (or some variant thereof) in more depth. This is the aim of Acuaviva's PhD project. Natural questions include: when is a Banach space constructed via the Bourgain-Delbaen method isomorphic to its Cartesian square? When is it isomorphic to its hyperplanes? (The space constructed by Argyros and Haydon fail both of these otherwise very "natural" properties.)