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This work studies the interplay between Green functions, the index ofdeterminacy of spectral measures and interior finite rank perturbations ofJacobi operators. The index of determinacy quantifies the stability ofuniqueness of solutions of the moment problem. We give results on the constancyof this index in terms of perturbations of the corresponding Jacobi operators.The permanence of the $N$-extremality of a measure is also studied. A measure$\mu$ is $N$-extremal when the polynomials are dense in $L_2(\mathbb{R},\mu)$.As a by-product, we give a characterization of the index in terms of cyclicvectors. We consider a new inverse problem for Jacobi operators in whichinformation on the place where the interior perturbation occurs is obtainedfrom the index of determinacy.

Stability of determinacy and inverse spectral problems for Jacobi operators