Executive Summary : | Beurling's theorem says that the invariant subspace of the unilateral shift operator $S(f)=zf$ on Hardy space $H^2$ has the form $uH^2$, where u is an inner function. Note that $K_u^2=H^2\ominus uH^2$ is the invariant subspace of the backward shift operator $S^*$ on $H^2$ . $K_u^2$ is called the model space. Let $P$ denote the orthogonal projection from $L^2$ onto $H^2$ and $P_u$ denote the orthogonal projection from $L^2$ onto $K_u^2$. For $\psi \in L^\infty$, the Toeplitz operator $T_\psi$ defined on $H^2$ by $$T_\psi g=P(\psi g),~g\in H^2.$$ For $\psi \in L^\infty$, Hankel operator $H_\psi$ defined on $H^2$ by $$H_\psi g=(I-P)(\psi g),~g\in H^2.$$ The compressions of Toeplitz operators on $K_u^2$ are called truncated Toeplitz operators, which are defined by $$A_\psi f=P_u(\psi f),~f\in K^2_u.$$ Xiaoyuan, Ran, Yixin and Yufeng gave necessary and sufficient condition that the defect operator $I-A^*_\varphi A_\varphi$ of truncated Toeplitz operator $A_\varphi$ for $\varphi \in K_u^2\cap L^\infty$ with $||\varphi||_\infty \leq 1$ is of finite-rank on the model space $K_u^2$. Xiaoyuan, Ran, Yixin, and Yufeng obtained necessary and sufficient condition that the defect operator $I-A^*_\varphi A_\varphi$ of truncated Toeplitz operator $A_\varphi$ for $\varphi \in K_u^2\cap L^\infty$ is compact on the model space $K_u^2$. Researchers would like to try necessary and sufficient conditions for defect operators of commuting pairs of shift operators on corresponding defect space. |