## Abstract

This paper investigates conditions under which an infinite matrix will be bounded as a linear operator between two weighted ℓ^{1} 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 ℓ^{1} spaces, for suitable weights. Necessary conditions and separate sufficient conditions for an infinite matrix to be bounded on some weighted ℓ^{1} 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 ℓ^{2} (Hilbert) space.

Original language | English |
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Pages (from-to) | 4689-4700 |

Number of pages | 12 |

Journal | Linear Algebra and Its Applications |

Volume | 438 |

Issue number | 12 |

DOIs | |

State | Published - Jun 15 2013 |

### Bibliographical note

Funding Information:Corresponding author. E-mail addresses: Joseph_Williams@umanitoba.ca (J.J. Williams), qiang.ye@uky.edu (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

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