M-Matrices over Infinite Dimensional Spaces

The intention here is to present an overview of some very recent results on three classes of operators, extending the corresponding matrix results. The relevant notions that are generalized here are that of a P-matrix, a Q-matrix, and an M-matrix. It is widely known (in the matrix case) that these notions coincide for Z-matrices. While we are not able to prove such a relationship between these classes of operators over Hilbert spaces, nevertheless, we are able to establish a relationship between Q-operators and M-operators, extending an analogous matrix result. It should be pointed out that, in any case, for P-operators, some interesting generalizations of results for P-matrices vis-a-vis invertibility of certain intervals of matrices have been obtained. These were proved by Rajesh Kannan and Sivakumar [92]. Since these are new, we include proofs for some of the important results. The last section considers a class of operators that are more general than M-operators. In particular, we review results relating to the nonnegativity of the Moore–Penrose inverse of Gram operators over Hilbert spaces, reporting the work of Kurmayya and Sivakumar [61] and Sivakumar [125]. These results find a place here is due to the reason that they extend the applicability of results for certain subclasses of M-matrices to infinite dimensional spaces.