A Banach space with few operators

Assuming the axiom (of set theory)V=L (explained below), we construct a Banach space with density character ℵ₁ such that every (linear bounded) operatorT fromB toB has the forma I+T ₁, whereI is the identity, andT ₁ has a separable range. The axiomV=L means that all the sets in the universe are in the classL of sets constructible from ordinals; in a sense this is the minimal universe. In fact, we make use of just one consequence of this axiom, ℵ₁ proved by Jensen, which is widely used by mathematical logicians.