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

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