Bounded and compact Toeplitz+Hankel matrices
We show that an infinite Toeplitz+Hankel matrix $T(\varphi ) + H(\psi )$ generates a bounded [compact] operator on $\ell ^p(\mathbb N _0)$ with $1\leq p\leq \infty $ if and only if both $T(\varphi )$ and $H(\psi )$ are bounded [compact]. We also give analogous characterizations for Toeplitz+Hankel operators acting on the reflexive Hardy spaces. In both cases, we provide an intrinsic characterization of bounded operators of Toeplitz+Hankel form similar to the Brown–Halmos theorem. In addition, we establish estimates for the norm and the essential norm of such operators.