For \(B_1\) and \(B_2\) two commuting linear (not necessarily bounded) operators on a Banach space, the question of when \((B_1-B_2)\) has a bounded inverse on \(X\) is of interest. For example, consider the abstract Cauchy problem \[ {{d}\over{dt}}u(t)= A(u(t))+ f(t) \qquad (0\leq t\leq T), \] where \(A\) is a linear operator on a Banach space \(W\), \(X= C([0,T],W)\), \(B_1\) is differentiation and \(B_2\) the lifting of \(A\) to \(X\). Solutions will be given by \(u=(B_1-B_2)^{-1}f\). Often, spectral conditions will guarantee a bounded inverse. In this paper, it is shown that, when \(B_1\) and \(B_2\) are commuting operators such that \(-B_1\) generates a bounded strongly continuous semigroup and \(B_2\) generates an exponentially decaying strongly continuous holomorphic semigroup, although \((B_1-B_2)^{-1}\) may not be bounded, it is true that \((B_1-B_2)^{-1}(B_1)^{-r}\) and \((B_1-B_2)^{-1}(-B_2)^{-r}\) are bounded operators on \(X\). Applications of this result to the abstract Cauchy problem are given for the initial condition \(u(0)=0\) and for the boundary condition \(u(0)=u(T)\).