Chandler Davis

More matrix forms of the arithmetic-geometric mean inequality

For arbitrary

Perturbation of spectral subspaces and solution of linear operator equations

n \times n

A Cauchy-Schwarz inequality for operators with applications

matrices A, B, X, and for every unitarily invariant norm, it is proved that

An extremal problem in Fourier analysis with applications to operator theory

2|||A^ * XB|||\leqq |||AA^ * X + XBB^ * |||

A bound for the spectral variation of a unitary operator

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Comparing a Matrix to its Off-Diagonal Part

Abstract Let A and B be normal operators on a Hilbert space. Let K A and K B be subsets of the complex plane, at distance at least δ from each other; let E be the spectral projector for A belonging to K A , and let F be the spectral projector for B belonging to K B . Our main results are estimates of the form δ ‖ EF ‖ c ‖ E ( A − B ) F ‖; in some special situations, the constant c is as low as 1. As an application, we prove, for an absolute constant d , that if the space is finite-dimensional and if A and B are normal with ‖;A − B‖ ⩽ ϵ d , then the spectrum of B can be obtained from that of A (multiplicities counted) by moving each eigenvalue by at most ϵ. Our main results have equivalent formulations as statements about the operator equation AQ - QB = S . Let A and B be normal operators on perhaps different Hilbert spaces. Assume σ ( A ) K A and σ ( B ) K B , where K A , K B , and δ are as before. Then we give estimates of the forms δ ‖ Q ‖⩽ c ‖ AQ − QB ‖.

Explicit functional calculus

Abstract For any unitarily invariant norm on Hilbert-space operators it is shown that for all operators A , B , X and positive real numbers r we have ||| |A∗XB| r ||| 2 ⩽ ||| |AA∗X| r ||| ||| |XBB∗| r ||| . Some consequences are then discussed. A simple proof is given for the fact that for positive operators A , B the function [ spr (A t B t )] 1 t is monotone in t on the positive half line.

Extending the Kantorovic̆ inequality to normal matrices

A certain minimal extrapolation problem for Fourier transforms is known to have consequences for the determination of best possible bounds in some problems in linear operator equations and in perturbation of operators. In this paper we estimate the value of the constant in the Fourier-transform problem, by an analytic reformulation.

An extremal problem for extensions of a sesquillinear form

Let A and B be unitary operator on a finite-dimensional space and assume . We show that the spectrum of B can be obtained from that of A(multiplicites counted)by moving each eigenvalue by at most ∊.

An interlacing theorem for eigenvalues of self-adjoint operators

Let \( \mathcal{O} \) be the operation which for any n x n complex matrix replaces all its diagonal entries by zeroes. For various matrix norms, we study max A \( \mid \mid \mid \mathcal{O}{\rm A}\mid \mid \mid / \mid \mid \mid {\rm A}\mid \mid \mid \). Upper and lower bounds are obtained, but they agree only for the c p norms with p = 1,2,∞. For these latter norms, the value of the maximum is also obtained with A restricted to the subset A ≥ 0.