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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1$\begingroup$ I am not sure I really understand the question (what do you mean by "permutation of set of eigenpairs"?), but the kind of example you might want to keep in mind is where $H = R^2$ and the two matrices $S$ and $T$ are of rank $1$, but with different eigenvectors. $\endgroup$– Martin HairerCommented Apr 8, 2014 at 3:08
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$\begingroup$ By "permutation" do you mean an operator that permutes the eigenvectors of S? $\endgroup$– Nate EldredgeCommented Apr 8, 2014 at 3:33
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$\begingroup$ Both Gaussians are non degenerate(just made the edit). @nate yes, I mean an operator R which permutes eigenvectors of S and the resulting Gaussian is N(0,SR). $\endgroup$– user47295Commented Apr 8, 2014 at 6:19
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$\begingroup$ I meant N(0,R'SR). $\endgroup$– user47295Commented Apr 16, 2014 at 13:09
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