Laurent–Jacobi Matrices and the Strong Hamburger Moment Problem

Abstract

Let ℒ be the linear space of the Laurent polynomials and suppose that <⋅, ⋅<ℒ is a positive-definite Hermitian inner product in ℒ with the additional property that \( \left\langle {f\left( z \right),g\left( z \right)} \right\rangle {/Mathcal L} = \left\langle {f\left( z \right)\overline {g\left( z \right)} ,1} \right\rangle {/Mathcal L} \). Starting from the five-term recurrence relation for orthogonal Laurent polynomials with respect to <⋅, ⋅<ℒ, we derive Laurent–Jacobi matrices \({/Mathcal J} \) and \({/Mathcal K} \) for the multiplication operator and its inverse in ℒ. These matrices are real and symmetric, and \({/Mathcal J} \) generates a symmetric operator in the Hilbert space l2 with natural basis { en }n = 0∞. We show that this operator has deficiency indices (0, 0) or (1, 1) and that every self-adjoint extension A in l2 has simple spectrum with generating vector e0. Let E be the spectral measure of A. Then the measure μe0 given by μe0(Ω) =<E(Ω) e0, e0< for all Borel sets Ω in \({/Mathcal R} \), satisfies \(\int_{/Mathcal R} {f\overline g {d\mu }_{e_0 } } = \left\langle {f,g} \right\rangle {/Mathcal L} \) forf,gℒ. In this way, we obtain a solution μe0 of the Strong Hamburger Moment Problem (SHMP) for which ℒ is dense in L2(μe0). Some results concerning the relation between the deficiency indices andthe set of all solutions of the SHMP are established. Finally, we give an analogue of a theorem by M. H. Stone which tells us which self-adjoint operators are generatedby a Laurent–Jacobi matrix with deficiency indices (0, 0).