Property \((\omega)\) and its perturbations. (English) Zbl 1524.47027
Summary: A Hilbert space operator \(T\) is said to have property \((\omega_1)\) if \(\sigma_a(T)\backslash\sigma_{aw}(T)\subseteq\pi_{00}(T)\), where \(\sigma_a(T)\) and \(\sigma_{aw}(T)\) denote the approximate point spectrum and the Weyl essential approximate point spectrum of \(T\) respectively, and \(\pi_{00}(T)=\{\lambda\in\mathrm{ iso }\sigma(T)\), \(0<\dim N(T-\lambda I)<\infty\}\). If \(\sigma_a(T)\backslash\sigma_{aw}(T)=\pi_{00}(T)\), we say \(T\) satisfies property \((\omega)\). In this note, we investigate the stability of the property \((\omega_1)\) and the property \((\omega)\) under compact perturbations, and we characterize those operators for which the property \((\omega_1)\) and the property \((\omega)\) are stable under compact perturbations.
MSC:
| 47B02 | Operators on Hilbert spaces (general) |
| 47A53 | (Semi-) Fredholm operators; index theories |
| 47A10 | Spectrum, resolvent |
| 47A55 | Perturbation theory of linear operators |
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