RECORD
The weighted Hilbert--Schmidt numerical radius
Functional Analysis · Submitted 28 Nov 2021
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Furthermore, we give a refinement of the triangle inequality for the Hilbert--Schmidt norm as follows: \begin{align*} {\|A+B\|}_{2} \leq \sqrt{2w_{_{({\|\!\cdot\!\|}_{2},\nu)}}^2\left(\begin{bmatrix} 0 & A \\ B^* & 0 \end{bmatrix}\right) - (1-2\nu)^2{\|A-B\|}_{2}^2} \leq {\|A\|}_{2} + {\|B\|}_{2}.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
Let $\mathbb{B}(\mathcal{H})$ be the algebra of all bounded linear operators on a Hilbert space $\mathcal{H}$ and let $N(\cdot)$ be a norm on $\mathbb{B}(\mathcal{H})$. For every $0\leq \nu \leq 1$, we introduce the $w_{_{(N,\nu)}}(A)$ as an extension of the classical numerical radius by \begin{align*} w_{_{(N,\nu)}}(A):= \displaystyle{\sup_{\theta \in \mathbb{R}}} N\left(\nu e^{i\theta}A + (1-\nu)e^{-i\theta}A^*\right) \end{align*} and investigate basic properties of this notion and prove inequalities involving it. In particular, when $N(\cdot)$ is the Hilbert--Schmidt norm ${\|\!\cdot\!\|}_{2}$, we present several the weighted Hilbert--Schmidt numerical radius inequalities for operator matrices. Furthermore, we give a refinement of the triangle inequality for the Hilbert--Schmidt norm as follows: \begin{align*} {\|A+B\|}_{2} \leq \sqrt{2w_{_{({\|\!\cdot\!\|}_{2},\nu)}}^2\left(\begin{bmatrix} 0 & A \\ B^* & 0 \end{bmatrix}\right) - (1-2\nu)^2{\|A-B\|}_{2}^2} \leq {\|A\|}_{2} + {\|B\|}_{2}. \end{align*} Our results extend some theorems due to F.~Kittaneh et al. (2019).
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $\mathbb{B}(\mathcal{H})$ be the algebra of all bounded linear operators on a Hilbert space $\mathcal{H}$ and let $N(\cdot)$ be a norm on $\mathbb{B}(\mathcal{H})$. | background 1.00 | 0.09 | 0.02 | asserted 0.76 |
| s1 | For every $0\leq \nu \leq 1$, we introduce the $w_{_{(N,\nu)}}(A)$ as an extension of the classical numerical radius by \begin{align*} w_{_{(N,\nu)}}(A):= \displaystyle{\sup_{\theta \in \mathbb{R}}} N\left(\nu e^{i\theta}A + (1-\nu)e^{-i\theta}A^*\right) \end{align*} and investigate basic properties of this notion and prove inequalities involving it. | contribution 0.99 | 0.95 | 0.20 | asserted 1.00 |
| s2 | In particular, when $N(\cdot)$ is the Hilbert--Schmidt norm ${\|\!\cdot\!\|}_{2}$, we present several the weighted Hilbert--Schmidt numerical radius inequalities for operator matrices. | contribution 0.83 | 0.94 | 0.19 | asserted 1.00 |
| s3 | Furthermore, we give a refinement of the triangle inequality for the Hilbert--Schmidt norm as follows: \begin{align*} {\|A+B\|}_{2} \leq \sqrt{2w_{_{({\|\!\cdot\!\|}_{2},\nu)}}^2\left(\begin{bmatrix} 0 & A \\ B^* & 0 \end{bmatrix}\right) - (1-2\nu)^2{\|A-B\|}_{2}^2} \leq {\|A\|}_{2} + {\|B\|}_{2}. | result 0.94 | 0.95 | 0.31 | asserted 1.00 |
| s4 | \end{align*} Our results extend some theorems due to F.~Kittaneh et al. (2019). | interpretation 0.52 | 0.48 | 0.11 | asserted 0.98 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original abstract
Let $\mathbb{B}(\mathcal{H})$ be the algebra of all bounded linear operators on a Hilbert space $\mathcal{H}$ and let $N(\cdot)$ be a norm on $\mathbb{B}(\mathcal{H})$. For every $0\leq \nu \leq 1$, we introduce the $w_{_{(N,\nu)}}(A)$ as an extension of the classical numerical radius by \begin{align*} w_{_{(N,\nu)}}(A):= \displaystyle{\sup_{\theta \in \mathbb{R}}} N\left(\nu e^{i\theta}A + (1-\nu)e^{-i\theta}A^*\right) \end{align*} and investigate basic properties of this notion and prove inequalities involving it. In particular, when $N(\cdot)$ is the Hilbert--Schmidt norm ${\|\!\cdot\!\|}_{2}$, we present several the weighted Hilbert--Schmidt numerical radius inequalities for operator matrices. Furthermore, we give a refinement of the triangle inequality for the Hilbert--Schmidt norm as follows: \begin{align*} {\|A+B\|}_{2} \leq \sqrt{2w_{_{({\|\!\cdot\!\|}_{2},\nu)}}^2\left(\begin{bmatrix} 0 & A \\ B^* & 0 \end{bmatrix}\right) - (1-2\nu)^2{\|A-B\|}_{2}^2} \leq {\|A\|}_{2} + {\|B\|}_{2}. \end{align*} Our results extend some theorems due to F.~Kittaneh et al. (2019).
Human review
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.