Hermitian positive definite Toeplitz matrices and Hessenberg matrices

In the context of orthogonal polynomials, an interesting class of Hermitian positive definite (HPD) matrices are those that are moment matrices with respect to a measure mu with support on the complex plane. In a more general framework, we establish a one-to-one correspondence between infinite upper Hessenberg matrices with positive subdiagonal and HPD matrices. In the particular case of an HPD Toeplitz matrix T, the properties and the description of its associated Hessenberg matrix in terms of the well-known recursion coefficients, and in the context of orthogonal polynomials in the unit circle, can be obtained using only an algebraical approach. We give some definition of Hessenberg matrices D(alpha) associated to a certain sequence (<mml:msub>alpha n)n=0 infinity</mml:msubsup>, and we characterize when such matrices are asymptotically Toeplitz.