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.

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.

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.

Maximum Principles for Normal Matrices

Ky Fan maximum principle is a well-known observation about traces of certain hermitian matrices. In this note, we derive a powerful extension of this claim. The extension is achieved in three ways. First, traces are replaced with norms of diagonal matrices, and any unitarily invariant norm can be used. Second, hermitian matrices are replaced by normal matrices, so the rule applies to a larger class of matrices. Third, diagonal entries can be replaced with eigenvalues and singular values. It is shown that the new maximum principle is closely related to the problem of approximating one matrix by another matrix of a lower rank.

Low-Rank Positive Approximants of Symmetric Matrices

Given a symmetric matrix X, we consider the problem of finding a low-rank positive approximant of X. That is, a symmetric positive semidefinite matrix, S, whose rank is smaller than a given positive integer, , which is nearest to X in a certain matrix norm. The problem is first solved with regard to four common norms: The Frobenius norm, the Schatten p-norm, the trace norm, and the spectral norm. Then the solution is extended to any unitarily invariant matrix norm. The proof is based on a subtle combination of Ky Fan dominance theorem, a modified pinching principle, and Mirsky minimum-norm theorem.

ON OPERATOR EQUATION A=A~*C

Let H be a complex Hilbert space, B（H） the set of bounded linear operators on H, C the complex field. For any A, A-1∈B（H）, the operator C=A*-1A is called polar-product operator of A in (1)The properties of C were studied in (1)In [2], we have used the polar-product to show the solvability of the operator equation λA2+μA*2=αA*A+βAA*（λ, μ, α, β∈C）, and given all its solutions. On discus-

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.