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By using Hadamard products we give some reasonable upper and lower bounds of Golden-Thompson type for ∥ e H 1 +…+ H m ∥, where H i ( i = 1, 2, …, m ) are arbitrary Hermitian matrices and ∥·∥ is an arbitrary unitarily invariant norm.

In matrix theory, majorization plays a significant role. For instance, majorization relations among eigenvalues and singular values of matrices produce a lot of norm inequalities and even matrix inequalities. This survey article is intended as a review of recent results in matrix theory related to majorization.

When A and B are n × n positive semi-definite matrices, and C is an n × n Hermitian matrix, the validity of a quadratic inequality (x∗Ax)12(x∗Bx)12 ⩾ ¦x∗Cx¦ is shown to be equivalent to the existence of an n × n unitary matrix W such that A12WB12 + B12W∗A12 = 2C. Some related inequalities are also discussed.

1. Basic properties of majorization. 2. Isotone maps and algebraic operations. 3. Double sub- and superstochasticity. 4. Doubly stochastic matrices. 5. Doubly stochastic matrices with minimum permanent. 6. Comparison of eigenvalues. 7. Doubly stochastic maps.

Through study of electrical network connections, Anderson and Duffin [2] introduced the concept of parallel sum of two Hermitian semidefinite matrices, and subsequently Anderson [l] defined a matrix operation, called shorted operation to a subspace, for each Hermitian semidefinite matrix. If A and B are impedance matrices of two resistive n-port networks, then their parallel sum A:B is the impedance matrix of the parallel connection. If ports are partitioned to a group of s ports and to the remaining group of n-s ports, then the shorted matrix A, to the subspace VI2 spanned by the former group is the impedance matrix of the network obtained by shorting the last n-s ports. Parallel addition and shorted operation can be defined on the class of all bounded positive linear operators on a Hilbert space and are of great interest from the point of view of operator theory. In fact, Anderson and Trapp [3] pushed through this program and studied fundamental properties of these operations and their interconnections. Our purpose in this paper is to give some characterizations of parallel addition and shorted operation among operations on the class of all positive operators on a Hilbert space. Theorem 1 will show that the series-parallel inequality and the transformer inequality are, in some sense, characteristic for parallel addition. In Theorem 2, shorted operation is recovered through commutativity with parallel addition, while Theorem 3 will characterize

For p > 1 we present reasonable nonnegative functions f ( t ) on [0,∞) such that in the space of matrices 0 ⩽ A , B and B ⩽ f ( A ) imply B p ⩽ A p .