The Jacobi operator J in the Hilbert space l 2 (N) is the operator whose matrix representation with respect to the canonical basis in l 2 (N) is a semi-infinite Jacobi matrix of the form 
       q 1 b 1 0 0 · · · b 1 q 2 b 2 0 · · · 0 b 2 q 3 b 3 0 0 b 3 q 4 . . . . . . . . . . . . . . .         ,(1.1)
where q n ∈ R and b n > 0 for any n ∈ N (see in [2] the definition of the matrix representation of an unbounded symmetric operator). J is closed by definition and it may be self-adjoint or have deficiency indices (1,1). In this work we deal with self-adjoint operators, so, if J = J * , we consider its self-adjoint extensions denoted J (g) , where g ∈ R ∪ {∞} (see Definition 1 a)). If J = J * we assume J (g) = J for all g ∈ R ∪ {∞} (see Definition 1 b)). The two spectra inverse problem for Jacobi operators J (g) takes as input **data** the spectra of two operators in a operator family obtained by perturbing J (g) in a certain way. The solution of the problem is the finding of the matrix (1.1) and the "boundary condition at infinity" g if necessary. The case of the operator family consisting of rank-one perturbations of a self-adjoint Jacobi operator has been amply studied in [8,12,13] and, in the more general setting of rank-one perturbations of J (g) , in [19,23]. Rank-one perturbations can be seen as a change of the "boundary condition at the origin" for the corresponding difference equation (see [19,Appendix]). We remark that the case of finite Jacobi matrices has also been thoroughly studied (see [5,6,9,11,14]).