Weighted Moore–Penrose Inverses Associated with Weighted Projections on Indefinite Inner Product Spaces

Abstract

Let H be a Hilbert C * -module, and let H M be the indefinite inner space induced by a self-adjointable and invertible operator M on H . Given weighted projections P and Q on H M , let S λ,k = (P Q) k − λ(Q P) k for a pair (k, λ), where k is a natural number and λ is a complex number. It is proved that P Q − Q P is weighted Moore-Penrose invertible if and only if S λ,k is weighted Moore-Penrose invertible for every pair (k, λ).

Developing Reverse Order Law for the Moore–Penrose Inverse with the Product of Three Linear Operators

“…Recently, Panigrahy and Mishra presented some properties for the weighted Moore-Penrose inverse for an arbitrary order tensor via the Einstein product [12]; they also proposed the expression for the Moore-Penrose inverse of the product of two tensors via the Einstein product [13,14]. Until now, some reverse order laws for the known generalized inverses have been widely studied and extended into a complex number field, a Hilbert space, or an associative ring (see [2,3,5,6,9,11,[15][16][17][19][20][21][22][23] and references therein).…”