A reflexive Banach space with few operators via greedy walks on countable ordinals

A non-separable reflexive Banach space is constructed such that every bounded operator is the sum of a scalar multiple of the identity and an operator with separable range. While as not as exceptional as the recently announced example where every bounded operator is the sum of a scalar multiple of the identity and a compact operator, it is noteworthy for its use of an uncountable well-ordered set and some deep but elementary results about walks down from one member of the set to another.