On Iohvidov's proofs of the Fischer-Frobenius theorem

Abstract

then jT_1 is a Hankel matrix for all Toeplitz matrices TnI' Thi s proced ure, howe ver, does not carry Hermitian ToepLitz matrices to Hermitian (i.e., real) Hankel matrices. The theore m of FischerFroebenius asserts that a class of tran sformation s exist each of which uniformly carries Toeplitz matrices to Hankel matrices in such a way that Hermitian Toeplitz matrices are carried to Hermitian Hankel matrices. Recently I. C. Iohvidov has publi shed three proofs of this result. One of these proofs is a direct but somewhat intricate calculation; it may be found on pages 211-213 of [1]1. A second proof, to be found on page 217 of [1] and also in [2], makes a preliminary reduction to the case of positive definite ToepLitz matrices, then takes advantage of a decomposition of definite Toeplitz matrices known from the theory of the trigonometric moment problem. The third proof, in [2], avoids the .reduction to the positive definite case, and uses instead a more co mplicated decomposition of Toeplitz matrices due to Iohvidov and Krein [3, p. 338]. The purpose of this paper is to give a short and direct proof of the Fischer-Frobe nius theorem. Our proof is based on a simple decomposition of arbitrary Toeplitz matrices, for which the proof is almost a triviality and whic h was apparently not noticed in [1] and [2]. See equation (3). Iohvidov's techniques then may be applied to (3) to produce the des ired result rapidly.