Search

Information is obtained about best approximation of a matrix by positive semidefinite ones, in trace norm. The situation is much different from approximation in spectral and Euclidean norms.

Let T be a Hilbert-space operator of class $$\left\| {S^{ - 1} TS} \right\| \leqslant 1\,\,\,\,and\,\,\left\| {S^{ - 1} } \right\| \cdot \left\| S \right\| \leqslant \max (1,\rho )$$ in the sense of Sz.-Nagy and Foias. Then there exists an invertible operator S such that ∥S−1TS∥≤1 and ∥S−1∥·∥S∥≤max(1,ρ). The following estimate is valid for the bilateral limit: $$\mathop {lim}\limits_{n \to \infty } \{ \left\| {T^n h} \right\|^2 + \left\| {T*^n h} \right\|^2 \} \leqslant max(2,\rho )\left\| h \right\|^2 .$$ Here the constants max(1,ρ) and max(2,ρ) are the best possible in their respective cases.

Though total positivity appears in various branches of mathematics, it is rather unfamiliar even to linear algebraists, when compared with positivity. With some unified methods we present a concise survey on totally positive matrices and related topics.

An investigation is presented of basic properties of fermion density matrices and their importance in quantum chemistry. An advanced study of properties of the first-order density matrices and the general properties of higher-order ones is presented. (R.E.U.)

A theory of the Kondo effect on the de Haas-van Alphen oscillations is presented. Results of our calculations may explain some characteristics left unsolved in the experiments on dilute Zn Mn and Cu Cr alloys.

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 .

A real-valued continuous function f(2) defined on a (finite or infinite) interval of the real line is said to be operator monotone if, for any pair of bounded selfadjoint operators A, B on an infinite dimensional Hilbert space, A > B implies f (A) > f(B). Here f(A) and f(B) are defined according to the usual functional calculus, and > refers to the order relation induced by the cone of positive (semi-definite) operators. Operator monotony is quite a restrictive condition. In fact, for p > 0 the function f(2): = 2 p is operator monotone on the half-line [0, c~) if and only if 0 1 the function f(2) = 2 p possesses some property akin to operator monotony. In this direction Furuta I-3] recently observed that if r > 0, p > 0, q > 1 satisfies (1 + 2r)q > p + 2r then A > B > 0 implies A tp § 2,)/q > (ArBPA r) l/~. If p = 2r and q = 2, the requirement is automatically satisfied, hence A p > (Ap/2BPAP/2)I/2 whenever A > B > 0. When A is invertible, this means that

Connections of resistive multiport electrical networks give rise to several questions concerning comparisons among certain functions of impedance matrices of branch networks. In the present paper, some matrix inequalities, related to symmetric function means of an n-tuple of positive definite matrices, are proved to answer a part of the conjectures of Anderson, Morley and Trapp.

Various silyl enol ethers were employed as quenchers for the living radical polymerization of methyl methacrylate with the RCl/RuCl2(PPh3)3/Al(Oi–Pr)3 initiating system. The most effective quencher was a silyl enol ether with an electron-donating phenyl group conjugated with its double bond [CH2C(OSiMe3)(4-MeOPh) (2a)] that afforded a halogen-free polymer with a ketone terminal at a high end functionality [Fn ∼ 1]. Such silyl compounds reacted with the growing radical generated from the dormant chloride terminal and the ruthenium complex to give the ketone terminal via the release of the silyl group along with the chlorine that originated from the dormant terminal. In contrast, less conjugated silyl enol ethers such as CH2C(OSiMe3)Me were less effective in quenching the polymerization. The reactivity of the silyl compounds to the poly(methyl methacrylate) radical can be explained by the reactivity of their double bonds, namely, the monomer reactivity ratios of their model vinyl monomers without the silyloxyl groups. The lifetime of the living polymer terminal was also estimated by the quenching reaction mediated with 2a. © 2000 John Wiley & Sons, Inc. J Polym Sci A: Polym Chem 38: 4735–4748, 2000

The effects of additives such as methanol, water, and radical scavengers (TEMPO and galvinoxyl) were studied in the living polymerization of methyl methacrylate (MMA) with the PhCOCHCl2/RuCl2(PPh3)...

A real-valued continuous function on an interval gives rise to a map via functional calculus from the convex set of Hermitian matrices all of whose eigenvalues belong to the interval. Since the subpace of Hermitian matrices is provided with the order structure induced by the cone of positive semidefinite matrices, one can consider convexity of this map. We will characterize its convexity by the following trace-inequalities: for . A related topic will be also discussed.

For the numerical solution of the hypersingular integral equation of a notched half-plane problem we propose collocation methods which look for an approximation of the derivative of the solution of the original equation. This derivative is the solution of a Cauchy singular integral equation with additional fixed singularities. We also give a solvability analysis of the original equation which motivates the suggested numerical methods.