The representation and approximation for the weighted Moore-Penrose inverse in Hilbert space.
Let \(X\), \(Y\) be Hilbert spaces and let \(T: X\to Y\) be a bounded linear operator with closed range. Let \(M: Y\to Y\) and \(N:X\to X\) be positive definite linear operators. Then \(T^+_{MN}\) is said to be the weighted Moore-Penrose inverse of \(T\) if it is the unique linear operator satisfying \(TT^+_{MN}T= T\), \(T^+_{MN} TT^+_{MN}= T^+_{MN}\), \((MTT^+_N)^*= MTT^+_{MN}\), \((NT^+_{MN})^*= NT^*_{MN}T\). When \(X= \mathbb{C}^n\), \(Y= \mathbb{C}^m\), \(M= I_m\), \(N= I_n\), the weighted Moore-Penrose inverse is reduced to the Moore-Penrose inverse \(T^+\). In the paper under review, the author presents a representation and an approximation for the weighted Moore-Penrose inverses in Hilbert spaces.
scientific article; zbMATH DE number 1861403