Suppose we have 2 nondegenerate 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)?
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This is clearly a question involving 2 Gaussian measures(Also, I asked this question)
Equivalence of Gaussian measures on Hilbert space
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