Suppose we have 2 Gaussian measures given by  N(0,T) and N(0,S) supported on a separable Hilbert space H. T and S are such that eigenbasis of S lies in the cameron martin space of N(0,T). Furthermore, let S and T have the same eigenvalues.
Does there exist a permutation of set of eigenpairs of S such that the resulting Gaussian measure(after the permutation) is absolutely continuous wrt N(0,T)?