On the p-numerical radii of Hilbert space operators

In this paper, we give new results for the p-numerical radii wp⋅ of Hilbert space operators. It is shown, among other inequalities, that if A is a Hilbert space operator, which belongs to the Schatten p-class, then wpp(A)≥wp/2p/2(A2)2p/2+A∗A+AA∗p/2p/22p+infθ∈RRe⁡(eiθA)pp−Im⁡(eiθA)pp2 and wpp(A)≤2p/2−2infθ∈RRe⁡((eiθA)2)p/2p/2+A∗A+AA∗p/2p/24 for 4≤p<∞. Also, wpp(A)≥wp/2p/2A24+A∗A+AA∗p/2p/22p/2+2+infθ∈RRe⁡(eiθA)pp−Im⁡(eiθA)pp2 and wpp(A)≤infθ∈RRe⁡((eiθA)2)p/2p/2+A∗A+AA∗p/2p/22p/2 for 2≤p≤ 4, where ⋅p is the Schatten p-norm. Applications of these inequalities to certain classes of operators are also given.