Let \(H\) and \(K\) be two separable Hilbert spaces over \(\mathbb{C}\) with \(\dim H\leq\infty\) and \(\dim K\leq\infty\), \(\mathcal A\) be the set of all linear operators of \(H\) into \(K\), \({\mathcal B}(H)\) be the Banach algebra of all bounded linear operators on \(H\), \({\mathfrak S}_ \infty\) be the Banach space of all bounded compact linear operators of \(H\) into \(K\) and \(\alpha\) be a symmetric norming function which defines a norm \(|\cdot|_ \alpha\) on a certain linear subspace \({\mathfrak S}_ \alpha\) of \({\mathfrak S}_ \infty\). Moreover, let \((\Omega,{\mathfrak A},\mu)\) be a positive measure space, \(N: \Omega\to {\mathcal B}(H)\) be such a measurable function that \(N(\omega)\geq0\) and \(\| N(\omega)\|_{B(H)}=1\) for \(\omega\in\Omega\), \({\mathcal E}_{p,\alpha}\) with \(p\in(0,\infty]\) be the set of all functions \(\phi: \Omega\to{\mathcal A}\) such that \(\phi(\omega)N(\omega)^{1/p}\in{\mathfrak S}_ \alpha\) for \(\omega\in\Omega\) and \(\phi N^{1/p}\) is measurable, \({\mathcal L}^ p_ \alpha(Nd\mu)\) be the set of all \(\phi\in{\mathcal E}_{p,\alpha}\) such that \(\int|\phi N^{1/p}|^ p_ \alpha d\mu<\infty\) and \(L^ p_ \alpha(Nd\mu)\) be the set of all \(p\)-equivalence classes of functions of \({\mathcal L}^ p_ \alpha(Nd\mu)\). Spaces \({\mathcal L}^ p_ \alpha(Nd\mu)\) and \(L^ \infty_ \alpha(Nd\mu)\) are defined separately as usual. The necessary and sufficient conditions that \(L_ \alpha^ p(Nd\mu)\subseteq L_ \alpha^ q(Nd\mu)\) and \(L^ p_ \alpha(Nd\mu)\subseteq L^ q_ \alpha(Nd\mu)\) for some \(p,q\in(0,\infty]\) with \(p<q\) has been given.