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

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 .