Abstract. As is known, the corona theorem is in general not true for a function $F \in H^{\infty }(L(H))$, where $L(H)$ is the space of bounded operators on an infinite dimensional separable Hilbert space $H$. Combined with a relatively compact range $F({\msbm D})$, the approximation property (AP), either in $H^{\infty }$ or in $L(H)$, provides functions satisfying the corona theorem, see [Vit]. Here we prove by counterexamples that these two methods are independent. We also give some new examples of subspaces of $L(H)$ and quotient spaces $H^{\infty }/ BH^{\infty }$ satisfying (AP). To finish, we give a version of the corona theorem for functions in the operator Nevanlinna class having a relatively compact range.

AMS Subject Classification (1991): 47A56, 47A20, 46B28, 30D55

Received April 16, 2002, and in revised form March 6, 2003. (Registered under 2934/2009.)