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PreprintCC BY 4.0arXiv

The L2-Cartesian Numerical Radii of Hilbert Space Operators

  • Wasim Audeh
  • Fuad Kittaneh
  • Hamid Reza Moradi
  • Satyajit Sahoo
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Abstract

Let ℋ be a complex Hilbert space, and let T=H+iK∈ℬ(ℋ) be the Cartesian decomposition of T, where H=H* and K=K*. We introduce the L2-Cartesian numerical radius and study its basic properties. We compare it with the classical numerical radius ω(T). In particular, we establish the sharp estimate $ \frac{1}{\sqrt{2}}ω(T)\le \wcart{2}(T)\le ω(T), $ and establish lower and upper bounds in terms of the Cartesian parts H,K. We also give an equality characterization and explicit sharpness examples. As a consequence, we obtain $\frac{1}{4}\left\|T^{*}T+TT^{*}\right\|\le \wcart{2}(T)^2\le ω(T)^2. $ More generally, we study the Lp-Cartesian numerical radius and prove the sharp comparison $ c_p\,ω(T)\le \wcart{p}(T)\le ω(T),\qquad 1\le p<\infty. $ Finally, we present a block-operator analogue of the classical refinement of the triangle inequality.

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Cite

BibTeXRIS
BibTeX
@misc{audeh2026l2cartesian,
  title = {{The $L^2$-Cartesian Numerical Radii of Hilbert Space Operators}},
  author = {Wasim Audeh and Fuad Kittaneh and Hamid Reza Moradi and Satyajit Sahoo},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.04844},
  eprint = {2610.04844},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.04844},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The L^2-Cartesian Numerical Radii of Hilbert Space Operators
AU  - Wasim Audeh
AU  - Fuad Kittaneh
AU  - Hamid Reza Moradi
AU  - Satyajit Sahoo
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2610.04844
UR  - https://arxiv.org/abs/2610.04844
AB  - Let ℋ be a complex Hilbert space, and let T=H+iK∈ℬ(ℋ) be the Cartesian decomposition of T, where H=H^* and K=K^*. We introduce the L^2-Cartesian numerical radius and study its basic properties. We compare it with the classical numerical radius ω(T). In particular, we establish the sharp estimate $ \frac{1}{\sqrt{2}}ω(T)\le \wcart{2}(T)\le ω(T), $ and establish lower and upper bounds in terms of the Cartesian parts H,K. We also give an equality characterization and explicit sharpness examples. As a consequence, we obtain $\frac{1}{4}\left\|T^{*}T+TT^{*}\right\|\le \wcart{2}(T)^2\le ω(T)^2. $ More generally, we study the L^p-Cartesian numerical radius and prove the sharp comparison $ c_p\,ω(T)\le \wcart{p}(T)\le ω(T),\qquad 1\le p<\infty. $ Finally, we present a block-operator analogue of the classical refinement of the triangle inequality.
ER  -