2023/05/28 by Bhunia, Pintu
#15A60 #47A12 #47A30 #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2305.17657
Several numerical radius inequalities are studied by developing an extension of the Buzano's inequality. It is shown that if T is a bounded linear operator on a complex Hilbert space, then wn(T) amp;\leqamp; \frac12n-1 w(Tn)+ ∑k=1n-1 \frac12k ‖Tk ‖ ‖T ‖n-k, for every positive integer n≥ 2. This is a non-trivial improvement of the classical inequality w(T)≤ ‖T‖. The above inequality gives an estimation for the numerical radius of the nilpotent operators, i.e., if Tn=0 for some least positive integer n≥ 2, then w(T) amp;\leqamp; (∑k=1n-1 \frac12k ‖Tk ‖ ‖T ‖n-k)1/n ≤ ( 1- \frac12n-1)1/n ‖T‖. Also, we deduce a reverse inequality for the numerical radius power inequality w(Tn)≤ wn(T). We show that if ‖T‖≤ 1, then wn(T) amp;\leqamp; \frac12n-1 w(Tn)+ 1- \frac12n-1, for every positive integer n≥ 2. This inequality is sharp.