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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 allx ∈ H    . In this paper, the asymmetric Putnam-Fuglede theorem for the pair( A , B )of power-bounded operators is proved when (i) A andB ∗are n-∗-paranormal operators (ii) A is a( n , k )    -quasi-∗-paranormal operator with reduced kernel andB ∗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 fork = 0 , 1such that the defect operatorD Tis Hilbert-Schmidt class, thenT ∈  C 10.

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