A note on self-dual cones in Hilbert spaces

Abstract

A result of Tam says that if a nonnegative matrix A has a nonnegative generalized inverse X (that is, X satisfies the equation AXA = A) then, A(R₊ⁿ) = R(A) ∩ R₊ᵐ and are simplicial (the image of the nonnegative orthant under an invertible linear map). Although in general, a simplicial cone need not be self-dual, there is another inner product with respect to which it is self-dual. The aim of this note to bring out an analoge of this in infinite dimensional separable Hilbert spaces, although there is no notion of simpliciality in such spaces.