The critical point and the p-norm of the Hilbert L-matrix

The Hilbert L-matrix As=[aij(s)], where aij(s)=1/(max⁡{i,j}+s) with i,j≥0, was introduced in [3]. As a surprising property, we showed that its 2-norm is constant for s≥s0, where the critical point s0 is unknown but relies in the interval (1/4,1/2). In this note, using some delicate calculations we s...

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Bibliographic Details
Published in:Linear algebra and its applications 2022-02, Vol.634, p.1-14
Main Authors: Bouthat, Ludovick, Mashreghi, Javad
Format: Article
Language:English
Subjects:
Critical point
Infinite matrices
Linear algebra
Lower bounds
Operator norm
Sequence spaces
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Summary:The Hilbert L-matrix As=[aij(s)], where aij(s)=1/(max⁡{i,j}+s) with i,j≥0, was introduced in [3]. As a surprising property, we showed that its 2-norm is constant for s≥s0, where the critical point s0 is unknown but relies in the interval (1/4,1/2). In this note, using some delicate calculations we sharpen this result by improving the upper and lower bounds of the interval surrounding s0. Moreover, we establish that the same property persists for the p-norm of As matrices.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2021.10.011