On the #≤-Order on the Set of Linear Bounded Operators in Banach Space

摘要

Let M_n(C) denote the space of square matrices of order n with complex coefficients. Definition 1. A matrix A ∈ M_n(C) has index k (Ind A = k) if rk A~k = rk A~(k+1) and k is the least natural number with such a property. Definition 2. The group inverse matrix A~# for A ∈ M_n(C) is the matrix satisfying the relations AA~#A = A, A~#AA~# = A~#,, AA~# = A~#A. For a given matrix A, the group inverse matrix A~# exists if and only if Ind A = 1, i.e., rk A = rk A~2 (see [1], [2]). Moreover, A~# either does not exist or is unique; see [3], [4]. For the properties of the group inverse matrix, see [5]-[7].