The essential norm, block sizes, and some Generalized Hilbert operators on lp
Abstract
We examine the generalized Hilbert (matrix) operators Hg,γ : (an) ↦ ∑n=1∞ (k/n)γ (gk an)/(n+k) on the ℓp spaces, 1<p<∞, where g=(gn) is a sequence and −1/p<γ<1−1/p. Given a partition of the natural numbers ⋃j Ij = ℕ, we encode g to our investigation via its ℓp-means Gj over the sets Ij. We prove that if Ij increases exponentially, the data (Gj) characterizes boundedness, but is insufficient to determine the exact value of the essential norm. Nonetheless, if the growth rate of (Ij) is subexponential, the corresponding data (Gj), given that the tail converge, is sufficient to determine the exact value, in which case we also calculate it. Moreover, it is shown that the boundedness of (Gj) is sufficient, but not necessary (and necessary, but not sufficient) to ensure Hg,γ is bounded if the underlying partition (Ij) increases subexponentially (and superexponentially, respectively). Using the dual operator, we also prove that the essential norm of Hg,γ is comparable with limsupj G(j) when (Ij) grows exponentially.
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BibTeX
@misc{norrbo2026essential,
title = {{The essential norm, block sizes, and some Generalized Hilbert operators on lp}},
author = {David Norrbo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13403},
eprint = {2609.13403},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13403},
}
RIS
TY - UNPB
TI - The essential norm, block sizes, and some Generalized Hilbert operators on lp
AU - David Norrbo
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.13403
UR - https://arxiv.org/abs/2609.13403
AB - We examine the generalized Hilbert (matrix) operators H_g,γ : (a_n) ↦ ∑_n=1^∞ (k/n)^γ (g_k a_n)/(n+k) on the ℓ^p spaces, 1<p<∞, where g=(g_n) is a sequence and −1/p<γ<1−1/p. Given a partition of the natural numbers ⋃_j I_j = ℕ, we encode g to our investigation via its ℓ^p-means G_j over the sets I_j. We prove that if I_j increases exponentially, the data (G_j) characterizes boundedness, but is insufficient to determine the exact value of the essential norm. Nonetheless, if the growth rate of (I_j) is subexponential, the corresponding data (G_j), given that the tail converge, is sufficient to determine the exact value, in which case we also calculate it. Moreover, it is shown that the boundedness of (G_j) is sufficient, but not necessary (and necessary, but not sufficient) to ensure H_g,γ is bounded if the underlying partition (I_j) increases subexponentially (and superexponentially, respectively). Using the dual operator, we also prove that the essential norm of H_g,γ is comparable with limsup_j G(j) when (I_j) grows exponentially.
ER -