$(α,β)$-A-Normal Operators in Semi-Hilbertian Spaces

Abstract

Let H be a Hilbert space and let A be a positive bounded operator on H. The semi-inner product u | v A := Au | v , u, v ∈ H induces a semi-norm . A on H. This makes H into a semi-Hilbertian space. In this paper we introduce and prove some proprieties of (α, β)-normal operators according to semi-Hilbertian space structures. Furthermore we state various inequalities between the A-operator norm and A-numerical radius of (α, β)-normal operators in semi Hilbertian spaces.