Polynomially Riesz elements

Abstract

A Banach algebra element a ∈ A is said to be “polynomially Riesz”, relative to the homomorphism T : A → B, if there exists a nonzero complex polynomial p(z) such that the image T p(a) ∈ B is quasinilpotent.     

Local bifurcation from characteristic values with finite multiplicity and its application to axisymmetric buckled states of a thin spherical shell

The purpose of this paper is to study bifurcation points of the equation T(v) = L(λ,v) + M(λ,v), (λ,v) ? Λ × D in Banach spaces, where for any fixed λ ? Λ, T, L(λ,·) are linear mappings and M(λ,·) is a nonlinear mapping of higher order, M(λ,0) = 0 for all λ ? Λ. We assume that λ is a characteristic value of the pair (T, L) such that the mapping T – L(λ ,·) is Fredholm with nullity p and index s, p > s ? 0. We shall find some sufficient conditions to show that (λ ,0) is a bifurcation point of the above equation. The results obtained will be used to consider bifurcation points of the axisymmetric buckling of a thin spherical shell subjected to a uniform compressive force consisting of a pair of coupled non-linear ordinary differential equations of second order.     

Imbedding conditions for λ-matrices

We say that A(λ) is λ-imbeddable in B(λ) whenever B(λ) is equivalent to a λ-matrix having A(λ) as a submatrix. In this paper we solve the problem of finding a necessary and sufficient condition for A(λ) to be λ-imbeddable in B(λ). The solution is given in terms of the invariant polynomials of A(λ) and B(λ). We also solve an analogous problem when A(λ) and B(λ) are required to be equivalent to regular λ-matrices. As a consequence we give a necessary and sufficient condition for the existence of a matrix B, over a field F, with prescribed similarity invariant polynomials and a prescribed principal submatrix A.     

Quadratic matrix polynomials with a parameter

In this paper we examine matrix polynomials of the form L(λ) = Aλ² + εBλ + C in which ε is a parameter and A, B, C are positive definite. This arises in a natural way in the study of damped vibrating systems. The main results are concerned with the generic case in which det L(λ) has at least 2n − 1 distinct zeros for all ε ϵ [0, ∞). The values of ε at which there is a multiple zero of det L(λ) are of major interest in this analysis. The dependence of first degree factors of L(λ) on ε is also discussed.     

The characteristic polynomial of a principal subpencil of a Hermitian matrix pencil

Given n-square Hermitian matrices A,B, let A_i,B_i denote the principal (n?1)- square submatrices of A,B, respectively, obtained by deleting row i and column i. Let μ, λ be independent indeterminates. The first main result of this paper is the characterization (for fixed i) of the polynomials representable as det(μA_i?λB_i) in terms of the polynomial det(μA?λB) and the elementary divisors, minimal indices, and inertial signatures of the pencil μA?λB. This result contains, as a special case, the classical interlacing relationship governing the eigenvalues of a principal sub- matrix of a Hermitian matrix. The second main result is the determination of the number of different values of i to which the characterization just described can be simultaneously applied.     

Spherical functions and immanants

Let λ be an irreducible character of S_n corresponding to the partition (r,s) of n. Let A be a positive semidefinite Hermitian n × n matrix. Let d_λ (A) and per(A) be the immanants corresponding to λ and to the trivial character of S_n , respectively. A proof of the inequality dλ(A)≤λ(id)per(A) is given.     

A conjecture involving generalized matrix functions

This paper resolves the following conjecture of R. Merris: Let d^G_λ be the generalized matrix function determined by a subgroup G of the symmetric group S_m and an irreducible complex character λ of G. If A, B, and A?B are m-square positive semidefinite hermitian m-square matrices and d^G_λ(A)=d^G_λ(B)≠0, then A=B.     

p-cyclic matrices and the symmetric successive overrelaxation method

In this paper, the new functional equation, [λ?(1?ω)²]^p=λ[λ+1?ω]^p-2(2?ω)²ω^pμ^p, which connects the eigenvalues μ of a particular weakly cyclic (of index p) Jacobi matrix B to the eigenvalues λ of its associated symmetric successive overrelaxation (SSOR) matrix S_ω, is derived. This functional equation is then applied to the problem of determining bounds for the intervals of convergence and divergence of the SSOR iterative method for classes of H-matrices.     

Normal series and character values in p-standard table algebras

We investigate the character values and structures of p-standard table algebras (A,B) with o(B) = p^N. If N≤3, then B has a complete normal series. If for every χ∈Irr(B), χ has at most p distinct classes of character values, and if either B has a complete normal series or p = 2, then B is an elementary abelian p-group.     

A construction for equidistant permutation arrays of index one

An equidistant permutation array (EPA) which we denote by A(r, λ; ν) is a ν × r array such that every row is a permutation of the integers 1, 2,…, r and such that every pair of distinct rows has precisely λ columns in common. R(r, λ) is the maximum ν such that there exists an A(r, λ; ν). In this paper we show that R(n² + n + 2, 1) ? 2n² + n where n is a prime power.     

Enumerating Groups Acting Regularly on a d-Dimensional Cube

Let K be a field of characteristic p > 0, K* the multiplicative group of K and G = G _p  × B a finite group, where G _p is a p-group and B is a p′-group. Denote by K ^λ G a twisted group algebra of G over K with a 2-cocycle λ ∈Z ²(G, K*). In this article, we give necessary and sufficient conditions for K ^λ G to be of OTP representation type, in the sense that every indecomposable K ^λ G-module is isomorphic to the outer tensor product V#W of an indecomposable K ^λ G _p -module V and an irreducible K ^λ B-module W.     

Generalized Path Spaces,II

A combination of the LIAPUNOV-SCHMIDT procedure, the implicit function theorems and the topological degree theory is used to investigate bifurcation points of equations of the form T(v) = L(λ, v) + M(λ, v), (λ, v) ? A × D?, where A is an open subset in a normed space and for every fixed λ ? A, T, L(λ ·) and M(λ ·) are mappings from the closure D? of a neighborhood D of the origin in a BANACH space X into another BANACH space Y with T(0) = L(λ, 0) = M(λ, 0) = 0. Let Λ be a characteristic value of the pair (T, L) such that T ? L( λ ,·) is a FREDHOLM mapping with nullity p and index s, p > s ≧ 0. Under suitable hypotheses on T. L and M, (λ , 0) is a bifurcation point of the above equations. This generalizes the results of [4], [6], [8], [13] and [14] etc. An application of the obtained results to the axisymmetric buckling problem of a thin spherical shell will be given.     

Enumerative Properties of NC (B) (p,q)

We determine the rank generating function, the zeta polynomial and the M?bius function for the poset NC ^(B) (p, q) of annular non-crossing partitions of type B, where p and q are two positive integers. We give an alternative treatment of some of these results in the case q = 1, for which this poset is a lattice. We also consider the general case of multiannular noncrossing partitions of type B, and prove that this reduces to the cases of non-crossing partitions of type B in the annulus and the disc.     

Simultaneous quasidiagonalization of complex matrices

Let

be the complex algebra generated by a pair of n × n Hermitian matrices A, B. A recent result of Watters states that A, B are simultaneously unitarily quasidiagonalizable [i.e., A and B are simultaneously unitarily similar to direct sums C₁⊕…⊕C_t,D₁⊕…⊕D_t for some t, where C_i, D_i are k_i × k_i and k_i?2(1?i?t)] if and only if [p(A, B), A]² and [p(A, B), B]² belong to the center of

for all polynomials p(x, y) in the noncommuting variables x, y. In this paper, we obtain a finite set of conditions which works. In particular we show that if A, B are positive semidefinite, then A, B are simultaneously quasidiagonalizable if (and only if) [A, B]², [A², B]² and [A, B²]² commute with A, B.     

A general theory of measures for nonnegative matrices

By a measure μ on the set N of m × n nonnegative matrices we mean that μ is a function from N to the nonnegative reals such that (i) μ(λA)=λμ(A) for all nonnegative λ and all A ∈ N, and (ii) μ(A + B) ? μ(A) for all A,B ? N. This paper develops a theory of such measures and shows how this theory can be applied to particular problems.     

Maps preserving operator pairs whose products are projections

Let B(H) be the algebra of all bounded linear operators on a complex Hilbert space H with dimH?2. It is proved that a surjective map φ on B(H) preserves operator pairs whose products are nonzero projections in both directions if and only if there is a unitary or an anti-unitary operator U on H such that φ(A)=λU^∗AU for all A in B(H) for some constants λ with λ²=1. Related results for surjective maps preserving operator pairs whose triple Jordan products are nonzero projections in both directions are also obtained. These show that the operator pairs whose products or triple Jordan products are nonzero projections are isometric invariants of B(H).     

A theorem of the alternatives for the equation Ax + B|x| = b

The following theorem is proved: given square matrices A, D of the same size, D nonnegative, then either the equation Ax?+?B|x|?=?b has a unique solution for each B with |B|?≤?D and for each b, or the equation Ax?+?B ₀|x|?=?0 has a nontrivial solution for some matrix B ₀ of a very special form, |B ₀|?≤?D; the two alternatives exclude each other. Some consequences of this result are drawn. In particular, we define a λ to be an absolute eigenvalue of A if |Ax|?=?λ|x| for some x?≠?0, and we prove that each square real matrix has an absolute eigenvalue.     

A graph which embeds all small graphs on any large set of vertices

For certain cardinals λ and κ a colouring P:[λ]²→λ is constructed such that if X ϵ[λ]^λ and Q:[κ]²→λ, then there is a one-to-one function i:κ→X such that P(i″A)=Q(A) for every Aϵ[κ]². Additional results are obtained.     

Differentiable Eigenvalues of Perturbed Quadratic Matrix Functions

Let T(λ, ε ) = λ² + λC + λεD + K be a perturbed quadratic matrix polynomial, where C, D, and K are n × n hermitian matrices. Let λ₀ be an eigenvalue of the unperturbed matrix polynomial T(λ, 0). With the falling part of the Newton diagram of det T(λ, ε), we find the number of differentiable eigenvalues. Some results are extended to the general case L(λ, ε) = λ² + λD(ε) + K, where D(ε) is an analytic hermitian matrix function. We show that if K is negative definite on Ker L(λ₀, 0), then every eigenvalue λ(ε) of L(λ, ε) near λ₀ is analytic.     

The relationship between Hadamard and conventional multiplication for positive definite matrices

It is known that if A is positive definite Hermitian, then A·A^-1⩾I in the positive semidefinite ordering. Our principal new result is a converse to this inequality: under certain weak regularity assumptions about a function F on the positive definite matrices, A·F(A)⩾AF(A) for all positive definite A if and only if F(A) is a positive multiple of A^-1. In addition to the inequality A·A^-1⩾I, it is known that A·A^-1T⩾I and, stronger, that λ_min(A·B)⩾λ_min(AB^T), for A, B positive definite Hermitian. We also show that λ_min(A·B)⩾λ_min(AB) and note that λ_min(AB) and λ_min(AB^T) can be quite different for A, B positive definite Hermitian. We utilize a simple technique for dealing with the Hadamard product, which relates it to the conventional product and which allows us to give especially simple proofs of the closure of the positive definites under Hadamard multiplication and of the inequalities mentioned.