The norm of an infinite L-matrix

Bouthat, Ludovick; Mashreghi, Javad

doi:10.7153/oam-2021-15-04

Cited by 9 publications

(5 citation statements)

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“…Finding a good lower bound for the number s 0 turned out to be harder than the upper bound, mainly due to the fact that we do not have a similar result to Theorem 2 from [1] for lower bounds. However, we can show that…”

Section: Approximating Smentioning

confidence: 83%

“…provided that 0 < ε < µ, where µ = µ(η) > 0. However, g(0) = 1−8s 2 2(4s−1) > 0 if 1 4 < s < s * ; thus we can set η = g(0) to be assured that there exist an ε small enough so that g(ε) > 0.…”

Section: Approximating Smentioning

confidence: 99%

“…acts as a bounded operator on ℓ 2 . This observation led to deeper study of C-matrices and L-matrices in [1] and [4], which are interesting subjects by themselves. More generally, it is often important to determine if an infinite matrix act as a bounded operator on some sequence space.…”

Section: Introductionmentioning

confidence: 99%

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The critical point and the $p$-norm of $A_s$ and $C$-matrices

Bouthat,

Mashreghi

2021

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The L-matrix A s = [1/(n+s)] was introduced in [1]. As a surprising property, we showed that its 2-norm is constant for s ≥ s 0 , where the critical point s 0 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 s 0 . Moreover, we show that the same property persists for the p-norm of A s matrices. We also obtain the 2-norm of a family of C-matrices with lacunary sequences.

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Section: Approximating Smentioning

confidence: 83%

Section: Approximating Smentioning

confidence: 99%

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The critical point and the $p$-norm of $A_s$ and $C$-matrices

Bouthat,

Mashreghi

2021

Preprint

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“…, a computation will show that L = CC * . The matrix above is a special case of an L-matrix and was explored recently in [6,7].…”

Section: The Cesàro Matrixmentioning

confidence: 99%

The Cesaro operator

Ross¹

2022

Preprint

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This paper surveys the various aspacts of the Cesàro operator with a special emphasis on the Hilbert space setting of ℓ 2 . We include a discussion of summability methods of Fourier series, the Cesàro matrix and integral operator, and the Cesàro operator on the Hilbert space ℓ 2 . In this setting of ℓ 2 we cover the spectral properties of the Cesàro operator as well as a treatment of its hyponormality (via posinormal operators) and subnormality (via a multiplication operator of Kriete and Trutt). Using the results of Kriete and Trutt, we describe the commutant of the Cesàro operator as well as its bounded square roots. The Cesàro operator has a rich lattice of invariant subspaces whose description remains unknown and we suspect might never be fully understood. We survey some results which display the complexity of these invariant subspaces and provide the reader with several paths forward for further discussion. Though the Cesàro operator was originally explored in the ℓ 2 and Hardy space settings, it has been explored in various other settings such as the general Hardy spaces as well as the Bergman spaces. Finally, we explore some of the so-called generalized Cesàro operator which are classes of integral operators on the Hardy spaces that connect to many areas of classical function theory.

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“…We use the notation T p for the norm of a linear operator from the sequence space p to itself. Several references have addressed the problem of finding the norm and lower bound of operators on matrix domains [1][2][3][4][5][6][7]. Our study considers infinite matrices [A] j,k , where all the indices j and k are non-negative.…”

Section: Introductionmentioning

confidence: 99%