2019/11/21 by Alomari, Mohammad W.
#FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.1912.01492
In this work, we improve and refine some numerical radius inequalities. In particular, for all Hilbert space operators T, the celebrated Kittaneh inequality reads: (1)/(4)‖ T^*T + TT^*‖≤ w2 (T ) ≤ (1)/(2)‖ T^*T + TT^*‖. In this work we provide some important refinements for the upper bound of the Kittaned inequality. Indeed, we establish w2 (T ) ≤ (1)/(2)‖ T^*T + TT^*‖ - (1)/(4) \mathop inf ‖ x ‖ = 1 ( ⟨ | T |x,x ⟩ - ⟨ | T^* |x,x ⟩ )2, which also refined and improved as w2 (T ) ≤ (1)/(2)‖ T^*T + TT^*‖ - (1)/(2) \mathop inf ‖ x ‖ = 1 ( ⟨ | T |x,x ⟩ - ⟨ | T^* |x,x ⟩ )2, and w2 (T ) ≤ (1)/(2) ‖T^*T+TT^* ‖ -(1)/(2) \mathop inf ‖ x ‖ = 1 (⟨ | T |2 x,x ⟩(1)/(2) - ⟨ | T^* |2 x,x ⟩(1)/(2))2, with third improvement w2 ( T ) ≤ (1)/(4 )‖ | T | + | T^* | ‖2 - (1)/(4 )\mathop inf ‖ x ‖ = 1 ( ⟨ | T | x,x ⟩ - ⟨ | T^* | x,x ⟩ )2. Other general related results are also considered.