Remark on the dilation of truncated Toeplitz operators

An operator Su & phi;,& psi; on L2 is called the dilation of a truncated Toeplitz operator if for two symbols & phi;, & psi; & ISIN; L & INFIN; and an inner function u, Su & phi;,& psi; f = & phi;Pu f +& psi;Quf holds for f & ISIN; L2 where Pu is the orthogonal projection of L2 onto Ku2 and Qu = I - Pu. In this paper, we study the squares of the dilation of truncated Toeplitz operators and the relation among its component operators. In particular, we provide characterizations for the square of the dilation of truncated Toeplitz operators Su & phi;,& psi; to be an isometry and a self-adjoint operator, respectively. As applications of the results, we find the cases where (Su & phi;,& psi;)2 is self-adjoint (resp., isometric) but Su & phi;,& psi; is not self-adjoint (resp., isometric).