A-numerical radius inequalities for semi-Hilbertian space operators

• Published in: Linear algebra and its applications Vol. 578; pp. 159 - 183
• Main Author: Zamani, Ali
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 01.10.2019; American Elsevier Company, Inc
• Subjects: A-adjoint operator; A-numerical radius; Commutators; Hilbert space; Inequality; Linear algebra; Lower bounds; Operators; Positive operator; Semi-inner product; Trigonometric functions; Upper bounds; secondary; Semi-inner product; A-numerical radius; A-adjoint operator; primary; Positive operator; Inequality
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2019.05.012

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.