2019/05/10 by Zamani, Ali · 4 citations
#46C05 #47A05 #47A12 #47B65 #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.1905.04081
Let A be a positive bounded operator on a Hilbert space (H, ⟨ ⋅, ⋅⟩ ). The semi-inner product ⟨ x, y⟩A := ⟨ Ax, y⟩, x, y\inH 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 ‖α\fracT + T\sharpA2 + β\fracT - T\sharpA2i‖A, where T\sharpA 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 (1)/(2)‖T‖A ≤ max\√1 - |cos|2AT, (√(2))/(2)\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.