Suppose X is a closed subspace of Z = ${({{Sigma}^{infty}}_{n=1}Z_{n})}_{p}$ (1 < p < ${infty}$, dim $Z_{n}$ < ${infty}$). We investigate an isometrically isomorphic embedding of L(X)/K(X) into L(X, Z)/K(X, Z), where L(X, Z) (resp. L(X)) is the space of the bounded linear operators from X to Z (resp. from X to X) and K(X, Z) (resp. K(X)) is the space of the compact linear operators from X to Z (resp. from X to X).