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Killip-Simon-classes of Jacobi matrices with essential spectrum on two symmetric and of SMP matrices on two arbitrary intervals

Jacobi matrices probably are the most classical object in spectral theory, while CMV matrices are a comparably fresh one, although they are related to a very classical topic, namely to orhtogonal polynomials on the unit circle (in the same way as Jacobi matrices are related to orhogonal polynomials on the real axis). We will discuss the third member of this family. Our matrices are generated by the orthonormal systems of functions related to the so-called Strong Moment Problem. For this reason we call them SMP matrices. For instance, one can describe the spectral sets of periodic SMP matrices. Similarly to the case of their counterparts, the description is given by means of conformal mappings on hyperbolic, in this case, comb domains. One can represent functional models associated with periodic and almost periodic SMP matrices. We are especially enthusiastic about the role, which such matrices can play in the Killip-Simon-problem related to Jacobi matrices with the essential spectrum on two arbitrary intervals. The parametric description of SMP matrices of the Killip-Simon-class with their essential spectrum on two arbitrary intervals is the main result of this paper.