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

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 : = ⟨ A x , 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 = (T 1 , ... , T d) is given by ω A , p (T) = sup ‖ x ‖ A = 1 ∑ k = 1 d | 〈 T k x , x 〉 A | p 1 p. Our aim in this paper is to establish several bounds involving ω A , 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). [ABSTRACT FROM AUTHOR]