A note on operator Banach spaces containing a complemented copy of c0

Abstract.

Let X and Y Banach spaces. Two new properties of operator Banach spaces are introduced. We call these properties “boundedly closed” and “d-boundedly closed”. Among other results, we prove the following one. Let ${\cal U}(X, Y)$ an operator Banach space containing a complemented copy of c ₀. Then we have: 1) If ${\cal U}(X, Y)$ is boundedly closed then Y contains a copy of c ₀. 2) If ${\cal U}(X, Y)$ is d-boundedly closed, then X ^* or Y contains a copy of c ₀.