Generalized $A$-numerical radius of operators and related inequalities
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866911984738369536 |
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| author | Bhunia, Pintu Feki, Kais Paul, Kallol |
| author_facet | Bhunia, Pintu Feki, Kais Paul, Kallol |
| contents | Let $A$ be a non-zero positive bounded linear operator on a complex Hilbert space $(\mathcal{H},\langle\cdot,\cdot\rangle)$. Let $ω_A(T)$ denote the $A$-numerical radius of an operator $T$ acting on the semi-Hilbert space $(\mathcal{H},\langle\cdot,\cdot\rangle_A)$, where $\langle x, y\rangle_{A} :=\langle Ax, y\rangle$ for all $x,y\in \mathcal{H}$. Let $N_A(\cdot)$ be a seminorm on the algebra of all $A$-bounded operators acting on $\mathcal{H}$ and let $T$ be an operator which admits $A$-adjoint. Then, we define the generalized $A$-numerical radius as $$ω_{N_A}(T)=\displaystyle{\sup_{θ\in \mathbb{R}}}\; N_A\left(\frac{e^{iθ}T+e^{-iθ}T^{\sharp_A}}{2}\right),$$ where $T^{\sharp_A}$ denotes a distinguished $A$-adjoint of $T$. We develop several generalized $A$-numerical radius inequalities from which follows the existing numerical radius and $A$-numerical radius inequalities. We also obtain bounds for generalized $A$-numerical radius of sum and product of operators. Finally, we study $ω_{N_A}(\cdot)$ in the setting of two particular seminorms $N_A(\cdot)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_02559 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Generalized $A$-numerical radius of operators and related inequalities Bhunia, Pintu Feki, Kais Paul, Kallol Functional Analysis 47A12, 46C05, 47A05 Let $A$ be a non-zero positive bounded linear operator on a complex Hilbert space $(\mathcal{H},\langle\cdot,\cdot\rangle)$. Let $ω_A(T)$ denote the $A$-numerical radius of an operator $T$ acting on the semi-Hilbert space $(\mathcal{H},\langle\cdot,\cdot\rangle_A)$, where $\langle x, y\rangle_{A} :=\langle Ax, y\rangle$ for all $x,y\in \mathcal{H}$. Let $N_A(\cdot)$ be a seminorm on the algebra of all $A$-bounded operators acting on $\mathcal{H}$ and let $T$ be an operator which admits $A$-adjoint. Then, we define the generalized $A$-numerical radius as $$ω_{N_A}(T)=\displaystyle{\sup_{θ\in \mathbb{R}}}\; N_A\left(\frac{e^{iθ}T+e^{-iθ}T^{\sharp_A}}{2}\right),$$ where $T^{\sharp_A}$ denotes a distinguished $A$-adjoint of $T$. We develop several generalized $A$-numerical radius inequalities from which follows the existing numerical radius and $A$-numerical radius inequalities. We also obtain bounds for generalized $A$-numerical radius of sum and product of operators. Finally, we study $ω_{N_A}(\cdot)$ in the setting of two particular seminorms $N_A(\cdot)$. |
| title | Generalized $A$-numerical radius of operators and related inequalities |
| topic | Functional Analysis 47A12, 46C05, 47A05 |
| url | https://arxiv.org/abs/2109.02559 |