LAUSR

Let $${\mathcal {H}}$$H be a Hilbert space and let A be a positive bounded operator on $${\mathcal {H}}$$H. The semi-inner product $$\langle u\;|\;v \rangle _A:=\langle Au\;|\;v\rangle ,\;\;u,v \in {\mathcal {H}}$$⟨u|v⟩A:=⟨Au|v⟩,u,v∈H induces a semi-norm $$\left\| .\;\right\| _A$$.A on $${\mathcal {H}}.$$H. This makes $${\mathcal {H}}$$H into a semi-Hilbertian space. In this paper, we introduce a new class of operators called $$(\alpha ,\beta )$$(α,β)-A-normal operators in semi-Hilbertian spaces. Some structural properties of this class of operators are established.