TY - JOUR
T1 - Algebras of Convolution Type Operators with Continuous Data do Not Always Contain All Rank One Operators
AU - Karlovich, Alexei
AU - Shargorodsky, Eugene
N1 - Funding Information:
This work was supported by national funds through the FCT ? Funda??o para a Ci?ncia e a Tecnologia, I.P. (Portuguese Foundation for Science and Technology) within the scope of the project UIDB/00297/2020 (Centro de Matem?tica e Aplica??es). We are grateful to Helena Mascarenhas, who asked the first author about a possibility of refinement of Theorem?1.1 contained in Theorem?5.2. We thank the anonymous referee for useful remarks.
Funding Information:
This work was supported by national funds through the FCT – Fundação para a Ciência e a Tecnologia, I.P. (Portuguese Foundation for Science and Technology) within the scope of the project UIDB/00297/2020 (Centro de Matemática e Aplicaçães).
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG.
Copyright:
Copyright 2021 Elsevier B.V., All rights reserved.
PY - 2021/4
Y1 - 2021/4
N2 - Let X(R) be a separable Banach function space such that the Hardy-Littlewood maximal operator is bounded on X(R) and on its associate space X′(R). The algebra CX(R˙) of continuous Fourier multipliers on X(R) is defined as the closure of the set of continuous functions of bounded variation on R˙ = R∪ { ∞} with respect to the multiplier norm. It was proved by C. Fernandes, Yu. Karlovich and the first author [11] that if the space X(R) is reflexive, then the ideal of compact operators is contained in the Banach algebra AX(R) generated by all multiplication operators aI by continuous functions a∈ C(R˙) and by all Fourier convolution operators W(b) with symbols b∈ CX(R˙). We show that there are separable and non-reflexive Banach function spaces X(R) such that the algebra AX(R) does not contain all rank one operators. In particular, this happens in the case of the Lorentz spaces Lp,1(R) with 1 < p< ∞.
AB - Let X(R) be a separable Banach function space such that the Hardy-Littlewood maximal operator is bounded on X(R) and on its associate space X′(R). The algebra CX(R˙) of continuous Fourier multipliers on X(R) is defined as the closure of the set of continuous functions of bounded variation on R˙ = R∪ { ∞} with respect to the multiplier norm. It was proved by C. Fernandes, Yu. Karlovich and the first author [11] that if the space X(R) is reflexive, then the ideal of compact operators is contained in the Banach algebra AX(R) generated by all multiplication operators aI by continuous functions a∈ C(R˙) and by all Fourier convolution operators W(b) with symbols b∈ CX(R˙). We show that there are separable and non-reflexive Banach function spaces X(R) such that the algebra AX(R) does not contain all rank one operators. In particular, this happens in the case of the Lorentz spaces Lp,1(R) with 1 < p< ∞.
KW - Algebra of convolution type operators
KW - Continuous Fourier multiplier
KW - Hardy-Littlewood maximal operator
KW - Lorentz space
KW - Rank one operator
KW - Separable Banach function space
UR - http://www.scopus.com/inward/record.url?scp=85103878782&partnerID=8YFLogxK
U2 - 10.1007/s00020-021-02631-x
DO - 10.1007/s00020-021-02631-x
M3 - Article
AN - SCOPUS:85103878782
SN - 0378-620X
VL - 93
SP - 1
EP - 24
JO - INTEGRAL EQUATIONS AND OPERATOR THEORY
JF - INTEGRAL EQUATIONS AND OPERATOR THEORY
IS - 2
M1 - 16
ER -