On disjoint range operators in a Hilbert space

Abstract

For a bounded linear operator M in a Hilbert space H, various relations among the ranges R(M), R(M * ), R(M + M * ) and the null spaces N (M), N (M * ) are considered from the point of view of their relations to the known classes of operators, such as EP, co-EP, weak-EP, GP, DR, or SR. Particular attention is paid to the range projectors of the operators M, M * and some further characteristics of these projectors are derived as well.

Operators with compatible ranges

“…The study of DR matrices, i.e. operators on arbitrary Hilbert spaces was conducted by Deng et al [10]. Among others, the authors in [10] studied the classes of operators described in the following definition.…”

“…Such A is called group-invertible, and the solution of this system is called the group inverse of A. In [10], the list of classes of operators to be studied contains the group-invertible operators as well. We did not include it in Definition 1.1 since this class is not defined through interrelation between the ranges of an operator and its adjoint.…”

“…where P ∈ B(R(T )) is the orthogonal projection with the range R(T ) ∩ R(T * ) and the null-space R(T )⊖R(T )∩R(T * ). Also, if R(T ) is closed and A and B defined as in (1), then AA * +BB * ∈ B(R(T )) is invertible, and as in [10] we denote ∆ = (AA * + BB * ) −1 .…”