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On \((n,m)\)-\(A\)-normal and \((n,m)\)-\(A\)-quasinormal semi-Hilbertian space operators. (English) Zbl 07547248

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 \(m\).

MSC:

47-XX Operator theory
54E40 Special maps on metric spaces
47B99 Special classes of linear operators
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