  In this paper we define and lay the groundwork for studying a novel matrix
ensemble: totally positive Hessenberg Toeplitz operators, denoted TPHT. This is
the intersection of two ensembles that have been significantly explored:
totally positive Hessenberg matrices (TPH) and Hessenberg Toeplitz matrices
(HT). TPHT has a rich linear algebraic and spectral structure that we describe.
Along the way we find some previously unnoticed connections between certain
Toeplitz normal forms for matrices and Lie theoretic interpretations. We also
numerically study the spectral asymptotics of TPH matrices via the TPHT
ensemble and use this to open a study of TPHT with random symbols.
