  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.
