Remark on the dilation of truncated Toeplitz operators

Abstract

An operator $S_{\varphi,\psi}^{u}$ on $L^2$ is called the {\it dilation of a truncated Toeplitz operator} if for two symbols $\varphi,\psi\in L^{\infty}$ and an inner function $u$, $$S_{\varphi,\psi}^{u}f=\varphi P_uf+\psi Q_uf$$ holds for $f\in {L}^{2}$ where $P_{u}$ is the orthogonal projection of $L^2$ onto ${\cal K}_{u}^2$ and $Q_u=I-P_u.$
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 $S_{\varphi,\psi}^{u}$ to be an isometry and a self-adjoint operator, respectively. As applications of the results, we find the cases where ${(S_{\varphi,\psi}^{u})}^2$ is self-adjoint (resp., isometry) but $S_{\varphi,\psi}^{u}$ is not self-adjoint (resp., isometry).