Lifting the solutions of a Toeplitz type equation; the semigroup case $X = T_sX{T_s}^*$

Abstract

A Toeplitz operator with respect to a contractive representation ${T_s}$ of an abelian semigroup $∑ $ in a Hilbert space H is an operator $X ∈ B(H)$ such that $X = T_sX{T_s}^*$ for all $s ∈ ∑ $. We show that if ${T_s}$ has a minimal isometric dilation ${U_s} ⊂ B(K)$, then Toeplitz operators can be obtained in a unique way as compressions of operators $Y ∈ B(K)$, called Toeplitz symbols, such that $Y = U_sY{U_s}^*$. This approach to lifting the Toeplitz equation $X = T_sX{T_s}^*$ is shown to be unitarily equivalent to the one proposed by Muhly in 1972. We use our approach to extend to this case a number of theorems about classical Toeplitz operators and, finally, we show that some classes of well-known operators like Wiener-Hopf operators Toeplitz operators in $H²(T^d)$ or Toeplitz operators in the sense of Murphy fall within the class studied in this paper. The main objective of this paper is to provide a general framework that, we hope, will be useful in order to extend to wider classes of operators some of the more recent and deep advances in the theory of Toeplitz operators.