Factorization of k -quasihyponormal operators

Let A be the class of all operators T on a Hilbert space H such that R ( T * k T ) , the range space of T * K T , is contained in R ( T * k + 1 ) , for a positive integer k . It has been shown that if T ϵ A , there exists a unique operator C T on H such that ( i ) T * k T = T * k + 1 C T ; ( i i ) ‖ C T ‖ 2 = inf { μ : μ ≥ 0 and ( T * k T ) ( T * k T ) * ≤ μ T * k + 1 T * k + 1 } ; ( i i i ) N ( C T ) = N ( T * k T ) and ( i v ) R ( C T ) ⫅ R ( T * k + 1 ) ¯ The main objective of this paper is to characterize k -quasihyponormal; normal, and self-adjoint operators T in A in terms of C T . Throughout the paper, unless stated otherwise, H will denote a complex Hilbert space and T an operator on H , i.e., a bounded linear transformation from H into H itself. For an operator T , we write R ( T ) and N ( T ) to denote the range space and the null space of T .