On $(n,m)$-$A$-normal and $(n,m)$-$A$-quasinormal semi-Hilbertian space operators

Abstract

The purpose of the paper is to introduce and study a new class of operators on semi-Hilbertian spaces, i.e. spaces generated by positive semi-definite sesquilinear forms. Let ${\mathcal H}$ be a Hilbert space and let $A$ be a positive bounded operator on ${\mathcal H}$. The semi-inner product $\langle h\mid k\rangle_A:=\langle Ah\mid k\rangle$, $h,k \in{\mathcal H}$, induces a semi-norm $\|{\cdot}\|_A$. This makes ${\mathcal H}$ into a semi-Hilbertian space. An operator $T\in{\mathcal B}_A({\mathcal H})$ is said to be $(n,m)$-$A$-normal if $[T^n,(T^{\sharp_A})^m]:=T^n(T^{\sharp_A})^m-(T^{\sharp_A})^mT^n=0$ for some positive integers $n$ and $m$.

Keywords

• semi-hilbertian space
• $a$-normal operator
• $(n,m)$-normal operator
• $(n,m)$-quasinormal operator