Bryan E. Cain

The inertia of a Hermitian matrix having prescribed diagonal blocks

Abstract For i =1,…, m let H i be an n i × n i Hermitian matrix with inertia In( H i )= ( π i , ν i , δ i ). We characterize in terms of the π i , ν i , δ i the range of In( H ) where H varies over all Hermitian matrices which have a block decomposition H = ( X ij ) i , j =1,…, m in which X ij is n i × n j and X ii = H i .

Multiplicative perturbations of stable and convergent operators

Abstract Some familiar classes of stable Hilbert-space operators are studied to determine how they overlap and where the unitary similarity classes of their members lie. Analogous, but less familiar, classes of convergent operators are examined with the same aim. The classes considered are often sets of products M A where M is a given set of diagonal or Hermitian matrices and A is a single matrix. The As for which M A is a set of stable or convergent operators are sometimes characterized.

The inertia of diagonal multiples of 3×3 real matrices

Abstract To each triple (α, β, γ) of non-negative integers satisfying α + β + γ = 3 there corresponds the class of 3×3 real matrices M such that the inertia In ( MD ) = ( α , β , γ ) for every 3×3 positive definite diagonal matrix D . Each such class is characterized by giving algebraic conditions which the principal minors of its members satisfy. These characterizations are obtained as corollaries of a general theorem on the roots of real homogeneous polynomials of order 3 and degree 3, and they make it possible to characterize for 3×3 matrices (1) those M such that In( MD ) = In ( D ) for all diagonal D and (2) those M such that MD is stable if and only if D is stable. The latter is the n = 3 case of the original definition of D -stability due to Arrow and McManus [1] and Enthoven and Arrow [3].

Theorems of the Alternative for Cones and Lyapunov Regularity of Matrices

Standard facts about separating linear functionals will be used to determine how two cones C and D and their duals C* and D* may overlap. When T: V → W is linear and K ⊂ V and D ⊂ W are cones, these results will be applied to C = T(K) and D, giving a unified treatment of several theorems of the alternate which explain when C contains an interior point of D. The case when V = W is the space H of n × n Hermitian matrices, D is the n × n positive semidefinite matrices, and T(X) = AX + X* A yields new and known results about the existence of block diagonal Xs satisfying the Lyapunov condition: T(X) is an interior point of D. For the same V, W and D, T(X) = X − B* XB will be studied for certain cones K of entry-wise nonnegative Xs.

Inequalities for monotonic arrangements of eigenvalues

Abstract This article corrects, clarifies, and extends results in [5] on inequalities for sequence rearrangements and for eigenvalues. The prototypes for these results are the inequalities of Hardy, Littlewood, and Polya about monotonic rearrangements. We examine some analogous results for eigenvalues of matrices and of their products.

Solutions to x−axb=w

Abstract Let T ( w )= awb , where a , b , w ∈ A , the bounded linear operators on a Hilbert space. We settle an open question of Redheffer and Redlinger by showing that the spectral radius of T may be 1 even though Σ T k ( w ) converges for every w ∈ A .

Convergent multiples of convergent operators

Abstract Let “ X ≫0” mean that “the bounded linear Hilbert-space operator X is selfadjoint, positive, and invertible”. We discuss the operators A which are known to be convergent (i.e. have spectral radius less than 1) because they all satisfy Steins condition P−A ∗ PA≫0 for a fixed P ≫0. We use the convexity of this set of As to show that when certain operators are in it (and hence convergent), others (often multiples) must be also. Our results generalize, and are modivated by, some results in [A. Bhaya, E. Kaszkurewicz, Linear Algebra Appl. 187 (1993) 87–104).

Operators which remain convergent when multiplied by certain Hermitian operators

Abstract Let A denote a bounded linear operator on a Hilbert space. We study here those As for which 1 exceeds the supremum of the spectral radius of the MA for M in three order intervals of Hermitian operators, [−I,I], [0,I] , and the invertible members of [0, I ].