Means for matrices and comparison of their norms
Fumio Hiai, Hideki Kosaki · Indiana University Mathematics Journal · 1999
Systematic and unified investigation is made for matrix means.Arithmetic, logarithmic, geometric, and harmonic means as well as natural one-parameter families of matrix means interpolating them are considered, and various comparison results for these means in arbitrary unitarily invariant norms are obtained.Introduction.For Hilbert space operators H, K, X with H, K ≥ 0, the operator norm inequalitywas first noticed by McIntosh ([28]).Then, in [7] Bhatia and Davis established the same inequality for an arbitrary unitarily invariant norm ||| • |||.This norm inequality is known as the matrix arithmetic-geometric mean inequality, and has been under active investigation in the recent years.(See [3], [6] for the subject matter and references.)As far as a proof is concerned, the finite-dimensional case (i.e., the matrix case) is essential.Moreover, thanks to the standard 2 × 2 matrix trick, one can reduce a proof to the case H = K.Then, by the unitary invariance, one could assume that a positive matrix H is diagonal (with eigenvalues λ 1 , λ 2 , ..., λ N ) and it is plain to seewhere • means the Hadamard product.In [20], [26] this approach was used, and here noticing the positive definiteness of multiplier matrices is crucial.In 1129