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) operator T from B to B has the form a I+T ₁, where I is the identity, and T ₁ has a separable range. The axiom V=L means that all the sets in the universe are in the class L 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.