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Keywords:
semi-Hilbertian space; $A$-normal operator; $(n,m)$-normal operator; $(n,m)$-quasinormal operator
semi-Hilbertian space; $A$-normal operator; $(n,m)$-normal operator; $(n,m)$-quasinormal operator
Summary:
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 $m$.
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