Let \(H_ 1\) and \(H_ 2\) be subspaces of a (not necessarily separable) Hilbert space H. Identify \(A\in L(H_ 1,H_ 2)\) by its extension to H such that \(A| H^{\perp}_ 1=0\). Let \({\mathcal J}\) be a closed 2-sided ideal in L(H) and let \(W_ 0\) be a nontrivial invariant subspace of \(A_ 0\in L(H)\). Also, let \(V_ 0\) be a subspace such that \(A_ 0V_ 0\subset V_ 0\subset W_ 0\). The pair \((W_ 0,A_ 0)\) is said to be stable in \({\mathcal J}\) relative to \(V_ 0\) if there exist a neighbourhood \(N_ 0\) of \(A_ 0\) in \(A_ 0+K\) and a \(C^ 1\)-function f: \(N_ 0\to K\) such that \(f(A_ 0)=0\) and \([1+f(A)]W_ 0\) is an invariant subspace of \(A_ 0\) for all \(A\in N_ 0\), where \(K={\mathcal J}\cap L(W_ 0\ominus V_ 0,W^{\perp}_ 0).\) A necessary condition for stability implies that \((W_ 0,A_ 0)\) is not stable if \(W_ 0\) and \(W^{\perp}_ 0\) are countably infinite dimensional and \(A_ 0\) is compact. Also, a sufficient condition results in an invariant subspace theorem for compact perturbations of certain Hermitian operators.