On some inequalities for the generalized joint numerical radius of semi-Hilbert space operators

Abstract

Let A be a positive (semidefinite) bounded linear operator on a complex Hilbert space (H, ⟨ · , · ⟩). The semi-inner product induced by A is defined by ⟨ x, y⟩ A: = ⟨ Ax, y⟩ for all x, y∈ H and defines a seminorm ‖ · ‖ A on H. This makes H into a semi-Hilbert space. For p∈ [1 , + ∞) , the generalized A-joint numerical radius of a d-tuple of operators T= (T1, … , Td) is given by ωA,p(T)=sup‖x‖A=1(∑k=1d|〈Tkx,x〉A|p)1p.Our aim in this paper is to establish several bounds involving ωA,p(·). In particular, under suitable conditions on the operators tuple T, we generalize the well-known inequalities due to Kittaneh (Studia Math 168(1):73–80, 2005).