On the Dilation of Truncated Toeplitz Operators

An operator (S_{varphi ,psi }^{u}in mathcal {L}(L^2)) is called the dilation of a truncated Toeplitz operator if for two symbols (varphi ,psi in L^{infty }) and an inner function u,

$$begin{aligned} S_{varphi ,psi }^{u}f=varphi P_uf+psi Q_uf end{aligned}$$

holds for (fin {L}^{2}) where (P_{u}) denotes the orthogonal projection of (L^2) onto the model space (mathcal { K}_{u}^2=H^2{ominus }{{u}H^2}) and (Q_u=I-P_u.) In this paper, we study properties of the dilation of truncated Toeplitz operators on (L^{2}). In particular, we provide conditions for the dilation of truncated Toeplitz operators to be normal. As some applications, we give several examples of such operators.