''If X is a Banach space, the algebra of bounded linear operators on X is denoted by O(X). Now let \((B^ 0,B^ 1)\) be a pair of Banach spaces continuously embedded in a linear topological space.'' ''A Banach space B will be called an intermediate space provided we have \(B^ 0\cap B^ 1\subseteq B\subseteq B^ 0+B^ 1,\) with continuous inclusions. Write \(I(B^ 0,B^ 1)\) to denote the space of linear transformations T which map \(B^ 0+B^ 1\) into itself, and which satisfy \(T|_{B^ j}\in O(B^ j)\), \(j=1,0\). An intermediate space B is said to be an interpolation space with respect to \((B^ 0,B^ 1)\) if \(T\in I(B^ 0,B^ 1)\) implies that \(T|_ B\in O(B).''\) The main result of the paper: Let \(1