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Symmetry 2019, 11(1), 64;

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

Department of Mathematics, King Khalid University, P. O. Box 9004, Abha, Saudi Arabia
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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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
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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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