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

Al Mohammady, Samir ; Ould Beinane, Sid Ahmed ; Ould Ahmed Mahmoud, Sid Ahmed

Mathematica Bohemica, Tome 147 (2022) no. 2, pp. 169-186

Résumé

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 $m$.

Détail

DOI : 10.21136/MB.2021.0167-19

Classification : 47B20, 47B50, 47B99, 54E40
Keywords: semi-Hilbertian space; $A$-normal operator; $(n, m)$-normal operator; $(n, m)$-quasinormal operator