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

Abstract

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.