On Maps Preserving Unitarily Invariant Norms of the Spectral Geometric Mean

We consider maps on positive definite cones of von Neumann algebras preserving unitarily invariant norms of the spectral geometric means. The main results concern Jordan *-isomorphisms between C*-algebras, and show that they are characterized by the preservation of unitarily invariant norms of those operations.

Some Properties of the Optimal Preconditioner and the Generalized Superoptimal Preconditioner

The optimal preconditioner and the superoptimal preconditioner were proposed in 1988 and 1992 respectively. They have been studied widely since then. Recently, Chen and Jin [6] extend the superoptimal preconditioner to a more general case by using the Moore-Penrose inverse. In this paper, we further study some useful properties of the optimal and the generalized superoptimal preconditioners. Several existing results are extended and new properties are developed.

ON MATRIX UNITARILY INVARIANT NORM CONDITION NUMBER

In this paper, the unitarily invariant norm \\.\\ on C-mxn is used. We first discuss the problem under what case, a rectangular matrix A has minimum condition number K(A) = \\A\\ \\A(+)\\, where A(+) designates the Moore-Penrose inverse of A; and under what condition, a square matrix A has minimum condition number for its eigenproblem? Then we consider the second problem, i.e., optimum of K(A) = \\A\\ \\A(-1)\\(2) in error estimation.

Singular Values of Sums of Positive Semidefinite Matrices

For positive real numbers a,b,a+b≤max{a+b1/2 a1/2,b+a1/2b1/2}.In this note,we generalize this fact to matrices by proving that for positive semidefinite matrices A and B of order n,for any c∈[-1,1]and j=1,2,…,n,sj(A+B)≤sj((A⊕B)+φc(A,B))≤sj(A+|B1/2A1/2|)⊕(B+|A1/2B1/2|),where sj(X)denotes the j-th largest singular value of X andφc(A,B):=1/2((1+c)|B1/2A1/2|(1-c)A1/2B1/2(1-c)B1/2A1/2(1+c)|A1/2B1/2|).This result sharpens some known result.Meanwhile,some related results are established.

Norm Inequalities for Positive Semidefinite Matrices

This paper aims to discuss some inequalities involving unitarily invariant norms and positive semidefinite matrices. By using properties of unitarily invariant norms, we obtain two inequities involving unitarily invariant norms and positive semidefinite matrices, which generalize the result obtained by Bhatia and Kittaneh.