Hardy-type norms of matrices
Abstract
We show that the function that assigns to each square matrix $A$ the geometric mean of $|Ax|$ over all unit vectors $x$ is a norm. The same holds for $L^p$ means with $0<p<1$. Some properties of these norms are proved, and others are conjectured.
Disclosure
“uality n1 tr AT B ≤ ∥A∥H 4 ∥B∥H 1 (A, B ∈ Rn×n ) which is conjectured to hold for n > 2 even though it fails for n = 2. Generative AI disclosure. The author used ChatGPT by OpenAI and Claude by An- thropic during all stages of this work. The final version of the text was written by the author, who is responsible for its correctness. 2. Preliminaries Let Rn+ = {x ∈ R”
PDF page 2
- Classification
- Drafting limited passages
- Multiplier
- 5
- Verified
Structural counts
Pages 10 pdf
Theorems 2 source
Lemmas 3 source
Propositions 2 source
Corollaries 0 source
Definitions 0 source
Displayed equations 54 source
Bibliography entries 9 source
Appendix pages 0 estimated
Count notes
- Source counts use the expanded primary TeX file main.tex.
- Appendix pages include the first PDF page with an explicit Appendix heading through the final page.