Putnam-Fuglede type theorem for class $ \mathcal{A}_k $ operators

Ahmed Bachir; Nawal Ali Sayyaf; Khursheed J. Ansari; Khalid Ouarghi; Ahmed Bachir; Nawal Ali Sayyaf; Khursheed J. Ansari; Khalid Ouarghi

doi:10.3934/math.2021241

AIMS Mathematics

2021, Volume 6, Issue 4: 4073-4082. doi: 10.3934/math.2021241

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Research article

Putnam-Fuglede type theorem for class $ \mathcal{A}_k $ operators

1.
Department of Mathematics, King Khalid University, P.O.Box 9004, Abha, Saudi Arabia
2.
Department of Mathematics, College of Science, University of Bisha, Bisha, Saudi Arabia

Received: 16 July 2020 Accepted: 10 November 2020 Published: 04 February 2021
MSC : 47B47, 47A30, 47B20

We will call $ U\in B(X) $ as an operator of class $ \mathcal{A}_k $ if for some integer $ k $, the following inequality is satisfied:

$ \vert U^{k+1}\vert^{\frac{2}{k+1}}\geq \vert U\vert^{2}. $

In the present article, some basic spectral properties of this class are given, also the asymmetric Putnam-Fuglede theorem and the range kernel orthogonality for class $ \mathcal{A}_k $ operators are proved.
- Putnam-Fuglede theorem,
- hyponormal operator,
- class $ \mathcal{A}_k $ operator
Citation: Ahmed Bachir, Nawal Ali Sayyaf, Khursheed J. Ansari, Khalid Ouarghi. Putnam-Fuglede type theorem for class $ \mathcal{A}_k $ operators[J]. AIMS Mathematics, 2021, 6(4): 4073-4082. doi: 10.3934/math.2021241

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Abstract

We will call $ U\in B(X) $ as an operator of class $ \mathcal{A}_k $ if for some integer $ k $, the following inequality is satisfied:

$ \vert U^{k+1}\vert^{\frac{2}{k+1}}\geq \vert U\vert^{2}. $

In the present article, some basic spectral properties of this class are given, also the asymmetric Putnam-Fuglede theorem and the range kernel orthogonality for class $ \mathcal{A}_k $ operators are proved.

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