Bounded and compact Toeplitz+Hankel matrices

TL;DR

This paper characterizes when infinite Toeplitz+Hankel matrices generate bounded or compact operators on sequence and Hardy spaces, providing intrinsic criteria and norm estimates for these operators.

Contribution

It offers new intrinsic characterizations of bounded and compact Toeplitz+Hankel operators on sequence and Hardy spaces, extending classical theorems.

Findings

Boundedness and compactness characterized by individual operators' properties.

Provides norm and essential norm estimates for Toeplitz+Hankel operators.

Extends Brown-Halmos theorem to Toeplitz+Hankel matrices.

Abstract

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.