On a new norm on $\mathcal{B}({\mathcal{H}})$ and its applications to numerical radius inequalities

Abstract

We introduce a new norm on the space of bounded linear operators on a complex Hilbert space, which generalizes the numerical radius norm, the usual operator norm and the modified Davis-Wielandt radius. We study basic properties of this norm, including the upper and the lower bounds for it. As an application of the present study, we estimate bounds for the numerical radius of bounded linear operators. We illustrate with examples that our results improve on some of the important existing numerical radius inequalities.2010 Mathematics Subject Classification. Primary 47A30, 47A12; Secondary 47A63. Key words and phrases. Numerical radius; bounded linear operator; operator inequalities; Hilbert space.Dr. Debmalya Sain feels elated to acknowledge the remarkable contribution of his beloved friend Subhajyoti Sarkar in his life. Mr. Pintu Bhunia would like to thank UGC, Govt. of India for the financial support in the form of SRF. Prof. Kallol Paul would like to thank RUSA 2.0, Jadavpur University for partial support.