On the isolated points of the operators satisfying absolute condition inequality. (English) Zbl 07825930
Summary: A \(m\)-quasi class \(A_k\), denoted as \(M\in\mathcal{Q}(A k,m)\), is defined as an operator \(M\) acting on a complex Hilbert space \(\mathcal{H}\), provided that the following condition holds true:
\[
M^{*m}|M^{k+1}|^{\frac{2}{k+1}}M^m \geq M^{*m}|M|^2 M^m,
\]
where both \(k\) and \(m\) are positive integers. In this research, we unveil fundamental structural characteristics of these operators. Leveraging these findings, we can establish that \(P\) is self-adjoint if \(P\) represents the Riesz idempotent associated with the isolated point \(\lambda\) in the spectrum of \(M\in\mathcal{Q}(A_k,m)\). Additionally, we provide a necessary and sufficient condition for \(M\otimes N\) to belong to \(\mathcal{QA}_k\) when both \(M\) and \(N\) are non-zero.
MSC:
| 47A55 | Perturbation theory of linear operators |
| 47A53 | (Semi-) Fredholm operators; index theories |
| 47B20 | Subnormal operators, hyponormal operators, etc. |
| 47A10 | Spectrum, resolvent |
| 47A11 | Local spectral properties of linear operators |
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