\((\alpha ,\beta )\)-A-Normal operators in semi-Hilbertian spaces

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