A-numerical radius inequalities for semi-Hilbertian space operators
Let A be a positive bounded operator on a Hilbert space (H,〈⋅,⋅〉). The semi-inner product 〈x,y〉A:=〈Ax,y〉, x,y∈H induces a semi-norm ‖⋅‖A on H. Let ‖T‖A and wA(T) denote the A-operator semi-norm and the A-numerical radius of an operator T in semi-Hilbertian space (H,‖⋅‖A), respectively. In this paper...
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| Published in | Linear algebra and its applications Vol. 578; pp. 159 - 183 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
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Amsterdam
Elsevier Inc
01.10.2019
American Elsevier Company, Inc |
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| Abstract | Let A be a positive bounded operator on a Hilbert space (H,〈⋅,⋅〉). The semi-inner product 〈x,y〉A:=〈Ax,y〉, x,y∈H induces a semi-norm ‖⋅‖A on H. Let ‖T‖A and wA(T) denote the A-operator semi-norm and the A-numerical radius of an operator T in semi-Hilbertian space (H,‖⋅‖A), respectively. In this paper, we prove the following characterization of wA(T)wA(T)=supα2+β2=1‖αT+T♯A2+βT−T♯A2i‖A, where T♯A is a distinguished A-adjoint operator of T. We then apply it to find upper and lower bounds for wA(T). In particular, we show that12‖T‖A≤max{1−|cos|A2T,22}wA(T)≤wA(T), where |cos|AT denotes the A-cosine of angle of T. Some upper bounds for the A-numerical radius of commutators, anticommutators, and products of semi-Hilbertian space operators are also given. |
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| AbstractList | Let A be a positive bounded operator on a Hilbert space (H,〈⋅,⋅〉). The semi-inner product 〈x,y〉A:=〈Ax,y〉, x,y∈H induces a semi-norm ‖⋅‖A on H. Let ‖T‖A and wA(T) denote the A-operator semi-norm and the A-numerical radius of an operator T in semi-Hilbertian space (H,‖⋅‖A), respectively. In this paper, we prove the following characterization of wA(T) wA(T)=supα2+β2=1‖αT+T♯A2+βT−T♯A2i‖A, where T♯A is a distinguished A-adjoint operator of T. We then apply it to find upper and lower bounds for wA(T). In particular, we show that 12‖T‖A≤max{1−|cos|A2T,22}wA(T)≤wA(T), where |cos|AT denotes the A-cosine of angle of T. Some upper bounds for the A-numerical radius of commutators, anticommutators, and products of semi-Hilbertian space operators are also given. Let A be a positive bounded operator on a Hilbert space (H,〈⋅,⋅〉). The semi-inner product 〈x,y〉A:=〈Ax,y〉, x,y∈H induces a semi-norm ‖⋅‖A on H. Let ‖T‖A and wA(T) denote the A-operator semi-norm and the A-numerical radius of an operator T in semi-Hilbertian space (H,‖⋅‖A), respectively. In this paper, we prove the following characterization of wA(T)wA(T)=supα2+β2=1‖αT+T♯A2+βT−T♯A2i‖A, where T♯A is a distinguished A-adjoint operator of T. We then apply it to find upper and lower bounds for wA(T). In particular, we show that12‖T‖A≤max{1−|cos|A2T,22}wA(T)≤wA(T), where |cos|AT denotes the A-cosine of angle of T. Some upper bounds for the A-numerical radius of commutators, anticommutators, and products of semi-Hilbertian space operators are also given. |
| Author | Zamani, Ali |
| Author_xml | – sequence: 1 givenname: Ali surname: Zamani fullname: Zamani, Ali email: zamani.ali85@yahoo.com organization: Department of Mathematics, Farhangian University, Tehran, Iran |
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| Snippet | Let A be a positive bounded operator on a Hilbert space (H,〈⋅,⋅〉). The semi-inner product 〈x,y〉A:=〈Ax,y〉, x,y∈H induces a semi-norm ‖⋅‖A on H. Let ‖T‖A and... |
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| SubjectTerms | A-adjoint operator A-numerical radius Commutators Hilbert space Inequality Linear algebra Lower bounds Operators Positive operator Semi-inner product Trigonometric functions Upper bounds |
| Title | A-numerical radius inequalities for semi-Hilbertian space operators |
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