Abstract

Let A be the class of all operators T on a Hilbert space H such that R(T*kT), the range space of T*KT, 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 CT on H such that (i)         T*kT=T*k+1CT ;(ii)        ‖CT‖2=inf{μ:μ≥0  and  (T*kT)(T*kT)*≤μT*k+1T*k+1} ;(iii)       N(CT)=N(T*kT) and(iv)       R(CT)⫅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 CT. 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.