Restricted Toeplitz and Hankel Operators

We introduce and systematically study a class of operators that arise naturally due to the Beurling decomposition of the Hardy space $H^2=K_\theta \oplus \theta H^2$. While the compressions of classical Toeplitz and Hankel operators to the Beurling subspace $\theta H^2$ and the model space $K_\theta$ account for the diagonal components of the decomposition, the corresponding off-diagonal operators have remained largely unexplored. Motivated by this, we introduce and analyze a new class of operators, termed \emph{restricted Toeplitz} and \emph{restricted Hankel operators}, acting between Beurling subspace $\eta H^2$ and model space $K_\theta$. Within this framework, we obtain necessary and sufficient conditions for the vanishing, finite-rank, and compactness properties of these operators. We further establish algebraic characterizations in the spirit of Brown-Halmos \cite{BH} and Sarason \cite{SAR, DES}, showing that these operators can be identified through certain operator equations involving compressed shifts. As an application, we introduce the notions of small and big truncated Toeplitz operators, and provide criteria for when they vanish, have finite rank, or are compact.