Laurent-Jacobi matrices and the strong hamburger moment problem

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 additiona l property that [f(z),g(z)](L) = [f(z)g((z) over bar), 1](L) for f, g is an element of L. Starting from the five-term recurrence relation for orthogon al Laurent polynomials with respect to [., .](L), we derive Laurent-Jacobi matrices J and K for the multiplication operator and its inverse in L. Thes e matrices are real and symmetric, and J generates a symmetric operator in the Hilbert space l(2) with natural basis {e(n)}(n=0)(infinity). We show th at this operator has deficiency indices (0, 0) or (1, 1) and that every sel f-adjoint extension A in l(2) has simple spectrum with generating vector e( 0). Let E be the spectral measure of A. Then the measure mu(e) given by mu( e)(Omega) = [E(Omega) e(0), e(0)] for all Borel sets Omega in R, satisfies integral(R) f (g) over bar d mu(e0) = [f, g](L) for f, g is an element of L . In this way, we obtain a solution mu(e) of the Strong Hamburger Moment Pr oblem (SHMP) for which L is dense in L-2(mu(e0)). Some results concerning t he relation between the deficiency indices and the 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 generated by a Laur ent-Jacobi matrix with deficiency indices (0, 0).