Bounded and compact Toeplitz+Hankel matrices

Abstract

We show that an infinite Toeplitz+Hankel matrix $T(\phi )+H(\psi )$ generates a bounded (compact) operator on ${\ell }^p({\mathbb{N}}_0)$ with $1\le p\le \infty$ if and only if both $T(\phi )$ 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.

21 pages; minor text fixes