Infinite matrices bounded on weighted ℓ1 spaces

Joseph J. Williams; Qiang Ye

doi:10.1016/j.laa.2013.02.023

Infinite matrices bounded on weighted ℓ¹ spaces

Joseph J. Williams, Qiang Ye

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

7 Scopus citations

Abstract

This paper investigates conditions under which an infinite matrix will be bounded as a linear operator between two weighted ℓ¹ spaces, and examines the relationship between the matrix and the weight vectors. It is shown that every infinite matrix is bounded as an operator between two weighted ℓ¹ spaces, for suitable weights. Necessary conditions and separate sufficient conditions for an infinite matrix to be bounded on some weighted ℓ¹ space (with the same weight for its domain and range) are given. We then show a connection between these results and the classical Schur Test which gives a sufficient condition for an infinite matrix to be bounded on the standard ℓ² (Hilbert) space.

Original language	English
Pages (from-to)	4689-4700
Number of pages	12
Journal	Linear Algebra and Its Applications
Volume	438
Issue number	12
DOIs	https://doi.org/10.1016/j.laa.2013.02.023
State	Published - Jun 15 2013

Bibliographical note

Funding Information:
Corresponding author. E-mail addresses: [email protected] (J.J. Williams), [email protected] (Q. Ye). 1 Research of this author was supported in part by NSF under Grant DMS-0915062.

Keywords

Infinite matrix
Matrix norm
Weighted sequence space
ℓ Space
ℓ Space

ASJC Scopus subject areas

Algebra and Number Theory
Numerical Analysis
Geometry and Topology
Discrete Mathematics and Combinatorics

Access to Document

10.1016/j.laa.2013.02.023

Cite this

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title = "Infinite matrices bounded on weighted ℓ1 spaces",

abstract = "This paper investigates conditions under which an infinite matrix will be bounded as a linear operator between two weighted ℓ1 spaces, and examines the relationship between the matrix and the weight vectors. It is shown that every infinite matrix is bounded as an operator between two weighted ℓ1 spaces, for suitable weights. Necessary conditions and separate sufficient conditions for an infinite matrix to be bounded on some weighted ℓ1 space (with the same weight for its domain and range) are given. We then show a connection between these results and the classical Schur Test which gives a sufficient condition for an infinite matrix to be bounded on the standard ℓ2 (Hilbert) space.",

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N2 - This paper investigates conditions under which an infinite matrix will be bounded as a linear operator between two weighted ℓ1 spaces, and examines the relationship between the matrix and the weight vectors. It is shown that every infinite matrix is bounded as an operator between two weighted ℓ1 spaces, for suitable weights. Necessary conditions and separate sufficient conditions for an infinite matrix to be bounded on some weighted ℓ1 space (with the same weight for its domain and range) are given. We then show a connection between these results and the classical Schur Test which gives a sufficient condition for an infinite matrix to be bounded on the standard ℓ2 (Hilbert) space.

AB - This paper investigates conditions under which an infinite matrix will be bounded as a linear operator between two weighted ℓ1 spaces, and examines the relationship between the matrix and the weight vectors. It is shown that every infinite matrix is bounded as an operator between two weighted ℓ1 spaces, for suitable weights. Necessary conditions and separate sufficient conditions for an infinite matrix to be bounded on some weighted ℓ1 space (with the same weight for its domain and range) are given. We then show a connection between these results and the classical Schur Test which gives a sufficient condition for an infinite matrix to be bounded on the standard ℓ2 (Hilbert) space.

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KW - Weighted sequence space

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