On (T,f)-connections of matrices and generalized inverses of linear operators

Abstract

In this note, generalized connections σ_{T ,f} are investigated, where Aσ_{T ,f} B = T_Af(T_A)^−(B) for positive semidefinite matrix A and hermitian matrix B, and operator monotone function f : J → R on an interval J ⊂ R. Here the symbol (T_A)^− denotes a reflexive generalized inverse of a positive bounded linear operator T_A. The problem of estimating a given generalized connection by other ones is studied. The obtained results are specified for special cases of α-arithmetic, α-geometric and α-harmonic operator means.