Invariance of stability properties of Hadamard and Szegِ product polynomials

It is shown that the integral representation of the Hadamard product operation of coefficient-wise multiplication of two polynomials, recently proved to keep the Hurwitz property invariant, provides interesting links to results in linear time-invariant discrete-time system theory. It is also pointed out that the invariance of the Schur property for stability of discrete-time systems, though known to be invalid under the Hadamard product operation, holds true under the operation composition. Various other related results on invariance are also included.