Structure of n-quasi left m-invertible and related classes of operators

Given Hilbert space operators , let and denote the elementary operators and . Let or . Assuming commutes with , and choosing to be the positive operator for some positive integer , this paper exploits properties of elementary operators to study the structure of -quasi -operators to bring together, and improve upon, extant results for a number of classes of operators, such as -quasi left -invertible operators, -quasi -isometric operators, -quasi -self-adjoint operators and -quasi symmetric operators (for some conjugation of ). It is proved that is the perturbation by a nilpotent of the direct sum of an operator satisfying , , with the 0 operator; if is also left invertible, then is similar to an operator such that . For power bounded and such that and , is polaroid (i.e., isolated points of the spectrum are poles). The product property, and the perturbation by a commuting nilpotent property, of operators satisfying , given certain commutativity properties, transfers to operators satisfying