Schatten class membership of Hankel operators on the unit sphere

Quanlei Fang; Jingbo Xia

doi:10.1016/j.jfa.2009.05.006

Schatten class membership of Hankel operators on the unit sphere

Quanlei Fang
, Jingbo Xia

Mathematics

University at Buffalo

SUNY Buffalo

Research output: Contribution to journal › Article › peer-review

23 Scopus citations

Abstract

Let H² (S) be the Hardy space on the unit sphere S in Cⁿ, n ≥ 2. Consider the Hankel operator H_f = (1 - P) M_f | H² (S), where the symbol function f is allowed to be arbitrary in L² (S, d σ). We show that for p > 2 n, H_f is in the Schatten class C_p if and only if f - P f belongs to the Besov space B_p. To be more precise, the "if" part of this statement is easy. The main result of the paper is the "only if" part. We also show that the membership H_f ∈ C_{2 n} implies f - P f = 0, i.e., H_f = 0.

Original language	English
Pages (from-to)	3082-3134
Number of pages	53
Journal	Journal of Functional Analysis
Volume	257
Issue number	10
DOIs	https://doi.org/10.1016/j.jfa.2009.05.006
State	Published - Nov 15 2009

Keywords

Hankel operator
Schatten class

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10.1016/j.jfa.2009.05.006

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title = "Schatten class membership of Hankel operators on the unit sphere",

abstract = "Let H2 (S) be the Hardy space on the unit sphere S in Cn, n ≥ 2. Consider the Hankel operator Hf = (1 - P) Mf | H2 (S), where the symbol function f is allowed to be arbitrary in L2 (S, d σ). We show that for p > 2 n, Hf is in the Schatten class Cp if and only if f - P f belongs to the Besov space Bp. To be more precise, the {"}if{"} part of this statement is easy. The main result of the paper is the {"}only if{"} part. We also show that the membership Hf ∈ C2 n implies f - P f = 0, i.e., Hf = 0.",

keywords = "Hankel operator, Schatten class",

author = "Quanlei Fang and Jingbo Xia",

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T1 - Schatten class membership of Hankel operators on the unit sphere

AU - Fang, Quanlei

AU - Xia, Jingbo

PY - 2009/11/15

Y1 - 2009/11/15

N2 - Let H2 (S) be the Hardy space on the unit sphere S in Cn, n ≥ 2. Consider the Hankel operator Hf = (1 - P) Mf | H2 (S), where the symbol function f is allowed to be arbitrary in L2 (S, d σ). We show that for p > 2 n, Hf is in the Schatten class Cp if and only if f - P f belongs to the Besov space Bp. To be more precise, the "if" part of this statement is easy. The main result of the paper is the "only if" part. We also show that the membership Hf ∈ C2 n implies f - P f = 0, i.e., Hf = 0.

AB - Let H2 (S) be the Hardy space on the unit sphere S in Cn, n ≥ 2. Consider the Hankel operator Hf = (1 - P) Mf | H2 (S), where the symbol function f is allowed to be arbitrary in L2 (S, d σ). We show that for p > 2 n, Hf is in the Schatten class Cp if and only if f - P f belongs to the Besov space Bp. To be more precise, the "if" part of this statement is easy. The main result of the paper is the "only if" part. We also show that the membership Hf ∈ C2 n implies f - P f = 0, i.e., Hf = 0.

KW - Hankel operator

KW - Schatten class

UR - https://www.scopus.com/pages/publications/70349459472

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DO - 10.1016/j.jfa.2009.05.006

M3 - Article

SN - 0022-1236

VL - 257

SP - 3082

EP - 3134

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 10

ER -

Schatten class membership of Hankel operators on the unit sphere

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Keywords

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