A note on quasi-paranormal operators. (English) Zbl 1291.47016
Let \(T\) be a bounded linear operator on an infinite-dimensional complex separable Hilbert space \({\mathcal H}\). In this paper, the authors consider the new class \({\mathcal Q}{\mathcal P}\) of operators satisfying \(\| T^2x\|^2\leq\| T^3x\|\,\| Tx\|\) for all \(x\in{\mathcal H}\) and study basic properties of such operators. In particular, they prove that, if \(T\in{\mathcal Q}{\mathcal P}\) and \(E\) is the Riesz idempotent for a nonzero isolated point \(\lambda_0\in\sigma(T)\), then \(E\) is self-adjoint if and only if \(N(T-\lambda_0)\subset N(T^*-\overline{\lambda_0})\).
Reviewer’s remark. In Proposition 2.13, saying that, if \(T\) is algebraically quasi-paranormal, then \(T\) has Bishop’s property \((\beta)\), the authors use the result from [K. Tanahashi and A. Uchiyama, Oper. Matrices 3, No. 4, 517–524 (2009; Zbl 1198.47039); doi:10.7153/oam-03-29] claiming that every paranormal operator has Bishop’s property \((\beta)\). As shown in [Oper. Matrices 7, No. 3, 737–738 (2013; Zbl 1291.47019), doi:10.7153/oam-07-42], this statement is in fact false.
Reviewer’s remark. In Proposition 2.13, saying that, if \(T\) is algebraically quasi-paranormal, then \(T\) has Bishop’s property \((\beta)\), the authors use the result from [K. Tanahashi and A. Uchiyama, Oper. Matrices 3, No. 4, 517–524 (2009; Zbl 1198.47039); doi:10.7153/oam-03-29] claiming that every paranormal operator has Bishop’s property \((\beta)\). As shown in [Oper. Matrices 7, No. 3, 737–738 (2013; Zbl 1291.47019), doi:10.7153/oam-07-42], this statement is in fact false.
Reviewer: Eungil Ko (Seoul)
MSC:
| 47B20 | Subnormal operators, hyponormal operators, etc. |
| 47A10 | Spectrum, resolvent |
| 47A11 | Local spectral properties of linear operators |
Keywords:
quasi-paranormal operator; algebraically quasi-paranormal operator; Bishop’s property \((\beta)\); SVEPReferences:
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