Some New Bounds on the Entropy Numbers of Diagonal Operators

Simon Fischer

doi:10.1016/j.jat.2019.105343

Some New Bounds on the Entropy Numbers of Diagonal Operators

Functional Analysis 2019-12-10 v3

Authors: Simon Fischer

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Abstract

Entropy numbers are an important tool for quantifying the compactness of operators. Besides establishing new upper bounds on the entropy numbers of diagonal operators $D_\sigma$ from $\ell_p$ to $\ell_q$ , where $p\not=q$ , we investigate the optimality of these bounds. In case of $p<q$ optimality is proven for fast decaying diagonal sequences, which include exponentially decreasing sequences. In case of $p>q$ we show optimality under weaker assumption than previously used in the literature. In addition, we illustrate the benefit of our results with examples not covered in the literature so far.

Keywords

entropy numbers diagonal operators upper bounds optimality compactness lp spaces

Cite

@article{arxiv.1903.00541,
  title  = {Some New Bounds on the Entropy Numbers of Diagonal Operators},
  author = {Simon Fischer},
  journal= {arXiv preprint arXiv:1903.00541},
  year   = {2019}
}

Comments

accepted manuscript in J.Approx.Theory

Some New Bounds on the Entropy Numbers of Diagonal Operators

Abstract

Keywords

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