IJPAM: Volume 95, No. 3 (2014)

THE SINGLE-VALUED EXTENSION PROPERTY AND
BISHOP'S PROPERTY (β) FOR CERTAIN
GENERALIZED CLASSES OF OPERATORS
ON HILBERT SPACES

Ould Ahmed Mahmoud Sid Ahmed
Department of Mathematics
College of Science
Al Jouf University
Al Jouf, 2014, KINGDOM OF SAUDI ARABIA


Abstract. An operator $T \in\mathcal{L}(\mathcal{H})$ is called of class $[nQN]$ if $T^n\big(T^*T\big)=\big(T^*T\big)T^n$ for a positive integer $n$, which is a common generalization of the quasi-normal and normal operators classes. Several properties of such class have been studied by the author in [22] and [23]. In this paper it is proved that in order to find a nontrivial subspace for a $n$-power quasi-normal operator $T$ it suffices to make the further assumption that $\sigma(T)\bigcap \sigma(\omega_k^{-1}T)= \emptyset$ where $\omega_k=\displaystyle\exp\big(i\frac{2k\pi}{n}\big)$ for $k=1,2,...,n-1$, (i.e., $\displaystyle\omega_k^n=1\bigg).$

Received: May 3, 2014

AMS Subject Classification: 47B20, 47B15

Key Words and Phrases: $n$-power quasi-normal,$m$-partial isometry-subscalar, single valued extension property (SVEP), Bishop's property, approximate spectrum, spectrum

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DOI: 10.12732/ijpam.v95i3.10 How to cite this paper?

Source:
International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2014
Volume: 95
Issue: 3
Pages: 427 - 452


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