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Symmetry 2019, 11(1), 64; https://doi.org/10.3390/sym11010064

Asymmetric Putnam-Fuglede Theorem for (n,k)-Quasi-∗-Paranormal Operators

1
Department of Mathematics, King Khalid University, P. O. Box 9004, Abha, Saudi Arabia
2
Department of Mathematics, Mascara University, Mascara 29000, Algeria
*
Author to whom correspondence should be addressed.
Received: 7 October 2018 / Revised: 27 November 2018 / Accepted: 29 November 2018 / Published: 8 January 2019
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Abstract

T B ( H ) is said to be ( n , k ) -quasi-∗-paranormal operator if, for non-negative integers k and n, T ( T k x ) ( 1 + n ) T ( 1 + n ) ( T k x ) T k x n ; for all x H . In this paper, the asymmetric Putnam-Fuglede theorem for the pair ( A , B ) of power-bounded operators is proved when (i) A and B are n-∗-paranormal operators (ii) A is a ( n , k ) -quasi-∗-paranormal operator with reduced kernel and B is n-∗-paranormal operator. The class of ( n , k ) -quasi-∗-paranormal operators properly contains the classes of n-∗-paranormal operators, ( 1 , k ) -quasi-∗-paranormal operators and k-quasi-∗-class A operators. As a consequence, it is showed that if T is a completely non-normal ( n , k ) -quasi-∗-paranormal operator for k = 0 , 1 such that the defect operator D T is Hilbert-Schmidt class, then T C 10 . View Full-Text
Keywords: Putnam-Fuglede theorem; hyponormal operator; (n,k)-quasi-∗-paranormal operator; paranormal operator; contraction; stable operator Putnam-Fuglede theorem; hyponormal operator; (n,k)-quasi-∗-paranormal operator; paranormal operator; contraction; stable operator
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Bachir, A.; Segres, A. Asymmetric Putnam-Fuglede Theorem for (n,k)-Quasi-∗-Paranormal Operators. Symmetry 2019, 11, 64.

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