Algebraic properties of Toeplitz + Hankel operators

Abstract

Recently an elegant result of Sang [29] characterizes when the product of two Toeplitz + Hankel operators is a Toeplitz + Hankel operator ((T+H)-operator). This result unifies several product problems for Toeplitz and Hankel operators. We extend the method of the author [19] on product problems for block Toeplitz and Hankel operators to give a short proof and some refinements of the result of Sang. Furthermore, we identify (T+H)-operators which are isometries or unitaries. We characterize when two arbitrary (T+H)-operators commute. We introduce several classes of (T+H)-operators which extend some classes of (T+H)-operators studied by Basor and Ehrhardt [2] [3] and Sang [29].