Further Results
One can see that the Hamburger moment problem is intimately related to orthogonal polynomials on the real line. The Gram–Schmidt procedure gives a basis of orthogonal polynomials in which the operator
has a tridiagonal Jacobi matrix representation. This in turn leads to a tridiagonal model of positive Hankel kernels.
An explicit calculation of the Cayley transform of T shows the connection with what is called the Nevanlinna class of analytic functions on the left half plane. Passing to the non-commutative setting, this motivates Krein's formula which parametrizes the extensions of partial isometries.
Read more about this topic: Hamburger Moment Problem
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