Nehari’s Theorem for Convex Domain Hankel and Toeplitz Operators in Several Variables

Karl-Mikael Perfekt,Marcus Carlsson

doi:10.1093/imrn/rnz193

Karl-Mikael Perfekt, Marcus Carlsson

Open Access

https://doi.org/10.1093/imrn/rnz193

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Abstract

Abstract We prove Nehari’s theorem for integral Hankel and Toeplitz operators on simple convex polytopes in several variables. A special case of the theorem, generalizing the boundedness criterion of the Hankel and Toeplitz operators on the Paley–Wiener space, reads as follows. Let $\Xi = (0,1)^d$ be a $d$-dimensional cube, and for a distribution $f$ on $2\Xi $, consider the Hankel operator $$\Gamma_f (g)(x)=\int_{\Xi} f(x+y) g(y) \, dy, \quad x \in\Xi.$$ Then $\Gamma _f$ extends to a bounded operator on $L^2(\Xi )$ if and only if there is a bounded function $b$ on ${{\mathbb{R}}}^d$ whose Fourier transform coincides with $f$ on $2\Xi $. This special case has an immediate application in matrix extension theory: every finite multilevel block Toeplitz matrix can be boundedly extended to an infinite multilevel block Toeplitz matrix. In particular, block Toeplitz operators with blocks that are themselves Toeplitz can be extended to bounded infinite block Toeplitz operators with Toeplitz blocks.

Highlights

For an open connected set ⊂ Rd, d ≥ 1, let= + = {x + y : x ∈, y ∈ }, Received December 7, 2018; Revised July 8, 2019; Accepted July 10, 20192 M
The case = R+ = (0, ∞) for d = 1 corresponds to the class of usual Hankel operators; when represented in the appropriate basis of L2(R+), the operator f,R+ is realized as an infinite Hankel matrix {an+m}∞ n,m=0 [31, Ch. 1.8]
Nehari’s Theorem in Several Variables 5 we prove Theorem 1.1, our Nehari theorem for Hankel operators

Summary

Introduction

For d > 1, the operators f ,Rd+ , = Rd+, correspond to (small) Hankel operators on the product domain multi-variable Hardy space Hd2 In this case, the analogue of Nehari’s theorem remains true, apart from (1.1), but it is significantly more difficult to prove. When d = 1, the only open connected sets ⊂ R are the intervals = I In this case, Theorem 1.1 is due to Rochberg [35], who called the corresponding operators f ,I Hankel/Toeplitz operators on the Paley–Wiener space. Every finite N × N d-multilevel block Toeplitz matrix can be extended to an infinite d-multilevel block Toeplitz matrix bounded on 2, with a constant that only depends on the dimension d.

Hankel operators on multi-variable Hardy spaces

Hankel operators on bounded domains

Toeplitz operators

Truncated correlation operators

Hankel operators on multi-variable Paley–Wiener spaces

Brief historical overview

Distribution symbols

On convex sets and polytopes

General domain Hankel operators

General domain Toeplitz operators