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

Abstract

Let A be a positive bounded linear operator acting on a complex Hilbert space H, • | • . Let ω A (T ) and T A denote the A-numerical radius and the A-operator seminorm of an operator T acting on the semi-Hilbertian space H, • | • A respectively, where x | y A := Ax | y for all x, y ∈ H. In this paper, we show that 1 4

Generalized Cauchy–Schwarz Inequalities and A-Numerical Radius Applications

“…When A = I in Equation ( 5), the resulting inequalities are the well established ones that were proven by Kittaneh in Theorem 1 in [18]. Conde et al in [19] established important numerical radius upper bounds. Specifically, for operators T, S ∈ L A (X ) and a positive integer n, the following inequalities hold:…”

“…To prove our first main result, we require three lemmas. We will establish the first one, while the second and third are quoted from the references [19,20], respectively. Let us start by presenting the first lemma, which concerns a refined version of the Cauchy-Schwarz inequality in the context of semi-Hilbert spaces.…”

Inequalities for the $A$-joint numerical radius of two operators and their applications

“…(3) The inequalities in (2.23) improve the bounds in (1.2) (see [18]). (4) A generalization of the inequalities in (2.23) are established in [11].…”

“…Further, the range and the kernel of T are denoted by R(T ) and N(T ), respectively. In addition, the cone of all positive operators on H is given by on H if and only if A is injective, and that (H, • A ) is complete if and only if R(A) is a closed subspace of H. For very recent contributions concerning operators acting on semi-Hilbert spaces, we refer the reader to [2,6,9,11] and the references therein. From now on, we suppose that A ∈ B(H) is always a positive (nonzero) operator and we denote the A-unit sphere of H by S A (0, 1), that is, S A (0, 1) := {x ∈ H ; x A = 1}.…”