Algebraic Properties of Generalized Right Invertible Operators

Abstract

ALGEBRAIC PROPERTIES OF GENERALIZED RIGHT INVERTIBLE OPERATORS Introduction [2]). However, the set of all generalized invertible operators is so large that, if we admit the axiom of choice, then every linear operator is generalized invertible [5]. Whereas, the generalized invertible operators do not satisfy desirable algebraic properties which the right invertible operators do (see [12]). For example, if a linear operator V € L(X) is generalized invertible and W is a generalized inverse of V, then there is not any general algorithm for constructing generalized inverses neither of V n , n € N, nor of algebraic polynomials induced by V. Hence, there is the lack of effective methods to solve equations induced by algebraic polynomials with a generalized invertible operator.In this paper, we introduce a so-called class of right invertible operators of degree r 6 N, and we denote its by R r (X). According to Definition 2, the set of all linear operators have been classified by means of the same invertible degree of operators (inclusion (2)). From classification (2) we can see that the set of all differential operators is exactly the first 1991 Mathematics Subject Classification: 34Gxx, 45A05, 45Exx. Key words and phrases: right invertible operators, algebraic operators, Volterra operators.