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Our aim in this article is to establish several new norm and numerical radius inequalities for products and sums of operators acting on a complex Hilbert space (H , 〈 · , · 〉) . Some of the obtained inequalities improve well known ones. In addition, by using new techniques, we prove certain new inequalities related to ω A (·) and | | · | | A , where ω A (T) and | | T | | A denote the A-numerical radius and the A-operator seminorm of an operator T acting on the semi-Hilbert space (H , 〈 · , · 〉 A) , respectively. Here 〈 x , y 〉 A : = 〈 A x , y 〉 for every x , y ∈ H . [ABSTRACT FROM AUTHOR]

The paper deals with the generalized numerical radius of linear operators acting on a complex Hilbert space H , which are bounded with respect to the seminorm induced by a positive operator A on H . Here A is not assumed to be invertible. Mainly, if we denote by ω A (⋅) and ω (⋅) the generalized and the classical numerical radii respectively, we prove that for every A-bounded operator T we have ω A (T) = ω ( A 1 / 2 T (A 1 / 2 ) † ) , where (A 1 / 2 ) † is the Moore-Penrose inverse of A 1 / 2 . In addition, several new inequalities involving ω A (⋅) for single and several operators are established. In particular, by using new techniques, we cover and improve some recent results due to Najafi [Linear Algebra Appl. 2020;588:489–496]. [ABSTRACT FROM AUTHOR]

In this paper, we obtain some refinements of the known operator inequalities for the p -Schatten class. In addition, we obtain an approach to the inequalities conjectured by Audenaert and Kittaneh for the p -Schatten class. [ABSTRACT FROM AUTHOR]