Anti-Hadamard matrices

An anti-Hadamard matrix may be loosely defined as a real (0, 1) matrix which is invertible, but only just. Let A be an invertible (0, 1) matrix with eigenvalues λ_i, singular values σ_i, and inverse B = (b_ij). We are interested in the four closely related problems of finding λ(n) = min_{A, i}|λ_i|, σ(n) = min_{A, i}σ_i, χ(n) = max_{A, i, j} |b_ij|, and μ(n) = max_AΣ_ijb²_ij. Then A is an anti-Hadamard matrix if it attains μ(n). We show that λ(n), σ(n) are between $\text{(2n)}{}^{\mathrm{?1}}\text{(}\text{n}\text{4}\text{)}{}^{\mathrm{<mtext>?n</mtext><mtext>2</mtext>}}$ and c√n (2.274)^?n, where c is a constant, $\text{c}\text{(2.274)}{}^{\mathrm{n}}\text{?χ(n)?2(}\text{n}\text{4}\text{)}{}^{\mathrm{<mtext>n</mtext><mtext>2</mtext>}}$, and $\text{c}\text{(5.172)}{}^{\mathrm{n}}\text{?μ(n)?4n}{}^2\text{(}\text{n}\text{4}\text{)}{}^{\mathrm{n}}$. We also consider these problems when A is restricted to be a Toeplitz, triangular, circulant, or (+1, ?1) matrix. Besides the obvious application—to finding the most ill-conditioned (0, 1) matrices—there are connections with weighing designs, number theory, and geometry.     

Best pseudo-isolated Gerschgorin disks for Eigenvalues

Let A be an arbitrary n×n matrix, partitioned so that if A=[A_ij], then all submatrices A_ii are square. If x is a positive vector, it is well-known that $\text{G(}\text{x}\text{) =∪}{}^{\mathrm{N}}{}_{\mathrm{i=1}}\text{G}{}_{\mathrm{i}}\text{(}\text{x}\text{)}$, where

$\text{G}{}_{\mathrm{i}}\text{(}\text{x}\text{) =}\text{z}\text{6(zI ? A}{}_{\mathrm{ii}}\text{)}{}^{\mathrm{?1}}\text{6}{}^{\mathrm{?1}}\text{?}\text{1}\text{x}{}_{\mathrm{i}}\text{∑}\text{j = 1}\text{j ≠ i}\text{N}\text{`6A}{}_{\mathrm{ij}}\text{6x}{}_{\mathrm{j}}$

, contains all the eigenvalues of A. The purpose of this paper is to give a new definition of the concept of an isolated subregion of G(x). An algorithm is given for obtaining the best such isolated subregion in a certain sense, and examples are given to show that tighter bounds for some eigenvalues of A may be obtained than with previous algorithms. For ease of computation, each subregion G_i(x) is replaced by the union of circular disks centered at the eigenvalues of A_ii.     

The asymptotic eigenvalue-distribution for a certain class of random matrices

For an n × n Hermitean matrix A with eigenvalues λ₁, …, λ_n the eigenvalue-distribution is defined by $\text{G(x, A) :=}\text{1}\text{n}$ · number {λ_i: λ_i ? x} for all real x. Let A_n for n = 1, 2, … be an n × n matrix, whose entries a_ik are for i, k = 1, …, n independent complex random variables on a probability space (Ω, $\text{R}$, p) with the same distribution F_a. Suppose that all moments $\text{E}$ | a | ^k, k = 1, 2, … are finite, $\text{E}$a=0 and $\text{E}$ | a | ². Let

$\text{M(A)=}\text{∑}\text{σ=1}\text{s}\text{θ}{}_{\mathrm{\sigma }}\text{P}{}_{\mathrm{\sigma }}\text{(A,A}{}^1\text{)}$

with complex numbers θ_σ and finite products P_σ of factors A and $\text{A}{}^1$ (= Hermitean conjugate) be a function which assigns to each matrix A an Hermitean matrix M(A). The following limit theorem is proved: There exists a distribution function G₀(x) = G₁x) + G₂(x), where G₁ is a step function and G₂ is absolutely continuous, such that with probability $\text{1 G(x, M(}\text{A}{}_{\mathrm{n}}\text{n}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{))}$ converges to G₀(x) as n → ∞ for all continuity points x of G₀. The density g of G₂ vanishes outside a finite interval. There are only finitely many jumps of G₁. Both, G₁ and G₂, can explicitly be expressed by means of a certain algebraic function f, which is determined by equations, which can easily be derived from the special form of M(A). This result is analogous to Wigner's semicircle theorem for symmetric random matrices (E. P. Wigner, Random matrices in physics, SIAM Review9 (1967), 1–23). The examples $\text{A}{}_{\mathrm{r}}\text{A}{}^{\mathrm{1r}}$, $\text{A}{}^{\mathrm{r}}\text{+ A}{}^{\mathrm{1r}}$, $\text{A}{}^{\mathrm{r}}\text{A}{}^{\mathrm{1r}}\text{± A}{}^{\mathrm{1r}}\text{A}{}^{\mathrm{r}}$, r = 1, 2, …, are discussed in more detail. Some inequalities for random matrices are derived. It turns out that with probability 1 the sharpened form

$\text{lim sup}\text{n→∞}\text{∑}\text{i=1}{}^{\mathrm{n}}\text{|γ}{}_{\mathrm{i}}{}^{\mathrm{(n)}}\text{|}{}^2\text{?6A}{}_{\mathrm{n}}\text{6}{}^2\text{? 0.8228…}$

of Schur's inequality for the eigenvalues λ_i⁽ⁿ⁾ of A_n holds. Consequently random matrices do not tend to be normal matrices for large n.     

On nonnormal matrices

For the eigenvalues λ_i of an n × n matrix A the inequality

$\text{∑}\text{i}\text{|λ}{}_{\mathrm{i}}\text{|}{}^2\text{(6A6}{}^4\text{?}\text{1}\text{2}\text{6D6}{}^2\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$

is proved, where $\text{D ? AA}{}^1\text{? A}{}^1\text{A}$ and 6 ● 6; denotes the euclidean norm. Conditions for equality are stated.     

Bounds for iterated normal matrices

Given a normal matrix A, asymptotic bounds are obtained for $\text{|}\text{A}{}^{\mathrm{m}}\text{|}{}_{\mathrm{\infty }}$ in terms of the spectral radius of A, the number of eigenvalues of A with modulus equal to the spectral radius of A, and the order of A. These results are extended to provide bounds for $\text{|}\text{A}{}^{\mathrm{m}}\text{|}{}_{\mathrm{\infty }}$ for all m ? 1.     

A fundamental theorem on lambda-matrices with applications —II. Difference equations with constant coefficients

The fundamental theorem of the title refers to a spectral resolution for the inverse of a lambda-matrix $\text{L(λ) =}\text{∑}\text{i=0}\text{l}\text{A}{}_{\mathrm{i}}\text{λ}{}^{\mathrm{i}}$ where the A_i are n×n complex matrices and detA_l ≠ 0. In this paper general solutions are formulated for difference equations of the form $\text{∑}\text{i=0}\text{l}\text{A}{}_{\mathrm{i}}\text{u}{}_{\mathrm{r\; +\; i}}\text{= ?}{}_{\mathrm{\gamma }}\text{, r = 1, 2,…}$. The use of these solutions is illustrated i new proof of Franklin's results describing the sums of powers of the eigenvalues of L(λ) (the generalized Newton identities), and in obtaining convergence proofs for the application of Bernoulli's method to the solution of $\text{∑}\text{i=0}\text{l}\text{A}{}_{\mathrm{i}}\text{S}{}^{\mathrm{i}}\text{= 0}$ for matrix S.     

Spectra of Cayley graphs

By a result of L. Lovász, the determination of the spectrum of any graph with transitive automorphism group easily reduces to that of some Cayley graph.We derive an expression for the spectrum of the Cayley graph X(G,H) in terms of irreducible characters of the group G:

for any natural number t, where ξ_i is an irreducible character (over $\text{C}$), of degree n_i , and λ_i,1 ,…, λ_i,ni are eigenvalues of X(G, H), each one n_i times. (σn_i² = n = | G | is the total'number of eigenvalues.) Using this formula for t = 1,…, n_i one can obtain a polynomial of degree n_i whose roots are λ_i,1,…,λ_i,ni. The results are formulated for directed graphs with colored edges. We apply the results to dihedral groups and prove the existence of k nonisomorphic Cayley graphs of D_p with the same spectrum provided p > 64k, prime.     

Eigenvalues of p-summing and lp-type operators in Banach spaces

This paper is a study of the distribution of eigenvalues of various classes of operators. In Section 1 we prove that the eigenvalues (λ_n(T)) of a p-absolutely summing operator, p ? 2, satisfy

$\text{∑}\text{n∈N}\text{|λ}{}_{\mathrm{n}}\text{(T)|}{}^{\mathrm{p}}{}^{\mathrm{<mtext>1</mtext><mtext>p</mtext>}}\text{?π}{}_{\mathrm{p}}\text{(T).}$

This solves a problem of A. Pietsch. We give applications of this to integral operators in L_p-spaces, weakly singular operators, and matrix inequalities.In Section 2 we introduce the quasinormed ideal Π₂⁽ⁿ⁾, P = (p₁, …, p_n) and show that for T ∈ Π₂⁽ⁿ⁾, 2 = (2, …, 2) ∈ Nⁿ, the eigenvalues of T satisfy

$\text{∑}\text{i∈N}\text{|λ}{}_{\mathrm{i}}\text{(T)|}{}^{\mathrm{<mtext>2</mtext><mtext>n</mtext>}}{}^{\mathrm{<mtext>n</mtext><mtext>2</mtext>}}\text{?π}{}^{\mathrm{n}}{}_2\text{(T).}$

More generally, we show that for T ∈ Π_p⁽ⁿ⁾, P = (p₁, …, p_n), pi ? 2, the eigenvalues are absolutely p-summable,

$\text{1}\text{p}\text{=}\text{∑}\text{i=1}\text{n}\text{1}\text{p}{}_{\mathrm{i}}\text{and}\text{∑}\text{n∈N}\text{|λ}{}_{\mathrm{n}}\text{(T)|}{}^{\mathrm{p}}{}^{\mathrm{<mtext>1</mtext><mtext>p</mtext>}}\text{?C}{}_{\mathrm{p}}\text{π}{}^{\mathrm{n}}{}_{\mathrm{P}}\text{(T).}$

We also consider the distribution of eigenvalues of p-nuclear operators on L_r-spaces.In Section 3 we prove the Banach space analog of the classical Weyl inequality, namely

$\text{∑}\text{n∈N}\text{|λ}{}_{\mathrm{n}}\text{(T)|}{}^{\mathrm{p}}\text{? C}{}_{\mathrm{p}}\text{∑}\text{n∈N}\text{α}{}_{\mathrm{n}}\text{(T)}{}^{\mathrm{p}}$

, 0 < p < ∞, where α_n denotes the Kolmogoroff, Gelfand of approximation numbers of the operator T. This solves a problem of Markus-Macaev.Finally we prove that Hilbert space is (isomorphically) the only Banach space X with the property that nuclear operators on X have absolutely summable eigenvalues. Using this result we show that if the nuclear operators on X are of type l₁ then X must be a Hilbert space.     

Extension de quelques theoremes sur les densites de series d'elements de n a des series de sous-ensembles finis de n

For a sequence A = {A_k} of finite subsets of N we introduce: $\text{δ(A) =}\text{inf}{}_{\mathrm{m?n}}\text{A(m)}\text{2}{}^{\mathrm{n}}$, $\text{d(A) =}\text{lim inf}{}_{\mathrm{n\to \infty }}\text{A(n)}\text{2}{}^{\mathrm{n}}$, where A(m) is the number of subsets A_k ? {1, 2, …, m}.The collection of all subsets of {1, …, n} together with the operation $\text{a ∪ b, (a ∩ b), (a 1 b = a ∪ b ? a ∩ b)}$ constitutes a finite semi-group N^∪ (semi-group N^∩) (group $\text{N}{}^1$). For N^∪, N^∩ we prove analogues of the Erdös-Landau theorem: δ(A+B) ? δ(A)(1+(2λ)^?1(1?δ(A>))), where B is a base of N of the average order λ. We prove for $\text{N}{}^{\mathrm{\cup }}\text{, N}{}^{\mathrm{\cap }}\text{, N}{}^1$ analogues of Schnirelmann's theorem (that δ(A) + δ(B) > 1 implies δ(A + B) = 1) and the inequalities λ ? 2h, where h is the order of the base.We introduce the concept of divisibility of subsets: a|b if b is a continuation of a. We prove an analog of the Davenport-Erdös theorem: if d(A) > 0, then there exists an infinite sequence {A_kr}, where A_kr | A_kr+1 for r = 1, 2, …. In Section 6 we consider for $\text{N∪, N∩, N1}$ analogues of Rohrbach inequality: $\text{2n}\text{? g(n) ? 2}\text{n}$, where g(n) = min k over the subsets {a₁ < … < a_k} ? {0, 1, 2, …, n}, such that every m? {0, 1, 2, …, n} can be expressed as m = a_i + a_j.Pour une série A = {A_k} de sous-ensembles finis de N on introduit les densités: $\text{δ(A) =}\text{inf}{}_{\mathrm{m?n}}\text{A(m)}\text{2}{}^{\mathrm{m}}$, $\text{d(A) =}\text{lim inf}{}_{\mathrm{n\to \infty }}\text{A(n)}\text{2}{}^{\mathrm{n}}$ où A(m) est le nombre d'ensembles A_k ? {1, 2, …, m}. L'ensemble de toutes les parties de {1, 2, …, n} devient, pour les opérations $\text{a ∪ b, a ∩ b, a 1 b = a ∪ b ? a ∩ b}$, un semi-groupe fini N^∪, N^∩ ou un groupe N¹ respectivement. Pour N^∪, N^∩ on démontre l'analogue du théorème de Erdös-Landau: δ(A + B) ? δ(A)(1 + (2λ)^?1(1?δ(A))), où B est une base de N d'ordre moyen λ. On démontre pour $\text{N}{}^{\mathrm{\cup }}\text{, N}{}^{\mathrm{\cap }}\text{, N}{}^1$ l'analogue du théorème de Schnirelmann (si δ(A) + δ(B) > 1, alors δ(A + B) = 1) et les inégalités λ ? 2h, où h est l'ordre de base. On introduit le rapport de divisibilité des enembles: a|b, si b est une continuation de a. On démontre l'analogue du théorème de Davenport-Erdös: si d(A) > 0, alors il existe une sous-série infinie {A_kr}, où A_kr|A_kr+1, pour r = 1, 2, … . Dans le Paragraphe 6 on envisage pour N^∪, $\text{N}{}^{\mathrm{\cap }}\text{, N}{}^1$ les analogues de l'inégalité de Rohrbach: $\text{2n}\text{? g(n) ? 2}\text{n}$, où g(n) = min k pour les ensembles {a₁ < … < a_k} ? {0, 1, 2, …, n} tels que pour tout m? {0, 1, 2, …, n} on a m = a_i + a_j.     

Uniqueness of solutions of nonlinear Dirichlet problems

In this paper, we consider the uniqueness of radial solutions of the nonlinear Dirichlet problem $\text{Δu + ?(u) = 0}\text{in}\text{Ω}\text{with}\text{u = 0}\text{on}\text{?Ω,}\text{where}\text{Δ = ∑}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{n}}\text{?}{}^2\text{?x}{}_{\mathrm{i}}{}^2\text{,?}$ satisfies some appropriate conditions and Ω is a bounded smooth domain in $\text{R}$ⁿ which possesses radial symmetry. Our uniqueness results apply to, for instance, $\text{?(u) = u}{}^{\mathrm{p}}\text{, p > 1}$, or more generally λu + ∑_{i = 1}^ka_iu^pi, λ ? 0, a_i > 0 and p_i > 1 with appropriate upper bounds, and Ω a ball or an annulus.     

Strong law for the bootstrap

Let X₁, X₂, X₃, … be i.i.d. r.v. with E|X₁| < ∞, E X₁ = μ. Given a realization X = (X₁,X₂,…) and integers n and m, construct Y_n,i, i = 1, 2, …, m as i.i.d. r.v. with conditional distribution $\text{P}{}^1\text{(Y}{}_{\mathrm{n,i}}\text{= X}{}_{\mathrm{j}}\text{) =}\text{1}\text{n}$ for 1 ? j ? n. ($\text{P}{}^1$ denotes conditional distribution given X). Conditions relating the growth rate of m with n and the moments of X₁ are given to ensure the almost sure convergence of $\text{(}\text{1}\text{m}\text{)Σ}{}^{\mathrm{m}}{}_{\mathrm{i=1}}\text{Y}{}_{\mathrm{n,i}}$ toμ. This equation is of some relevance in the theory of Bootstrap as developed by Efron (1979) and Bickel and Freedman (1981).     

On majorization and Schur products

Suppose A, D₁,…,D_m are n × n matrices where A is self-adjoint, and let $\text{X = Σ}{}^{\mathrm{m}}{}_{\mathrm{k\; =\; 1}}\text{D}{}_{\mathrm{k}}\text{AD}{}^1{}_{\mathrm{k}}$. It is shown that if $\text{ΣD}{}_{\mathrm{k}}\text{D}{}^1{}_{\mathrm{k}}\text{= ΣD}{}^1{}_{\mathrm{k}}\text{D}{}_{\mathrm{k}}\text{= I}$, then the spectrum of X is majorized by the spectrum of A. In general, without assuming any condition on D₁,…,D_m, a result is obtained in terms of weak majorization. If each D_k is a diagonal matrix, then X is equal to the Schur (entrywise) product of A with a positive semidefinite matrix. Thus the results are applicable to spectra of Schur products of positive semidefinite matrices. If A, B are self-adjoint with B positive semidefinite and if b_ii = 1 for each i, it follows that the spectrum of the Schur product of A and B is majorized by that of A. A stronger version of a conjecture due to Marshall and Olkin is also proved.     

Generalizations of two inequalities involving hermitian forms

Let λ₁ and λ_N be, respectively, the greatest and smallest eigenvalues of an N×N hermitian matrix H=(h_ij), and x=(x₁,x₂,…,x_N) with (x,x)=1. Then, it is known that (1) λ₁?(x,Hx)?λ_N and (2) if, in addition, H is positive definite, $\text{(λ}{}_1\text{+λ}{}_{\mathrm{N}}\text{)}{}^2\text{4λ}{}_1\text{λ}{}_{\mathrm{N}}\text{?(x,Hx)(x,H}{}^{\mathrm{?1}}\text{x)?1}$. Assuming that y=(y₁,y₂,…, y_N) and |y_i|?1, i=1,2,…,N, it is shown in this paper that these inequalities remain true if H and H^?1 are, respectively, replaced by the Hadamard products $\text{M(y)1H}$ and $\text{M(y)1H}{}^{\mathrm{?1}}$, where M(y) is a matrix defined by $\text{M(y)=(δ}{}_{\mathrm{ij}}\text{+(1?δ}{}_{\mathrm{ij}}\text{)}\text{y}{}_{\mathrm{i}}\text{y}{}_{\mathrm{j}}$. Subsequently, these results are extended to improve the spectral bounds of $\text{M(y)1H}$.     

Deterministic version of Wigner's semicircle law for the distribution of matrix eigenvalues

It is proved that Wigner's semicircle law for the distribution of eigenvalues of random matrices, which is important in the statistical theory of energy levels of heavy nuclei, possesses the following completely deterministic version. Let A_n=(a_ij), 1?i, ?n, be the nth section of an infinite Hermitian matrix, {λ⁽ⁿ⁾}_1?k?n its eigenvalues, and {u_k⁽ⁿ⁾}_1?k?n the corresponding (orthonormalized column) eigenvectors. Let $\text{v}{}^1{}_{\mathrm{n}}\text{=(a}{}_{\mathrm{n1}}\text{,a}{}_{\mathrm{n2}}\text{,?,a}{}_{\mathrm{n,n?1}}\text{)}$, put

$\text{X}{}_{\mathrm{n}}\text{(t)=[n(n-1)]}{}^{\mathrm{<mtext>-1</mtext><mtext>2</mtext>}}\text{∑}\text{k=1}\text{[(n-1)t]}\text{|v}{}_{\mathrm{n}}{}^1\text{u}{}_{\mathrm{f}}{}^{\mathrm{(n-1)}}\text{|}{}^2\text{,}\text{0?t?1}$

(bookeeping function for the length of the projections of the new row v¹_n of A_n onto the eigenvectors of the preceding matrix A_n?1), and let finally

$\text{F}{}_{\mathrm{n}}\text{(x)=n}{}^{-1}\text{(}\text{number of}\text{λ}{}_{\mathrm{k}}{}^{\mathrm{(n)}}\text{?x}\text{n}\text{,1?k?n)}$

(empirical distribution function of the eigenvalues of $\text{A}{}_{\mathrm{n}}\text{n}$. Suppose (i) $\text{lim}{}_{\mathrm{n}}\text{a}{}_{\mathrm{nn}}\text{n}\text{=0}$, (ii) lim_nX_n(t)=Ct(0<C<∞,0?t?1). Then

$\text{F}{}_{\mathrm{n}}\text{?W(·,C)}\text{(n→∞)}$

,where W is absolutely continuous with (semicircle) density

$\text{w(x,C)=}\text{(2Cπ)}{}^{-1}\text{(4C-x}{}^2{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{for}\text{|x|?2}\text{C}\text{0}\text{for}\text{|x|?2}\text{C}$

     

Infinite-variate wide-sense Markov processes and functional analysis for bounded operator-forming vectors

Let p, q be arbitrary parameter sets, and let $\text{H}$ be a Hilbert space. We say that x = (xⁱ)_i?q, xⁱ ? $\text{H}$, is a bounded operator-forming vector (?$\text{H}$_F^q) if the Gram matrix 〈x, x〉 = [(xⁱ, x^j)]_i?q,j?q is the matrix of a bounded (necessarily ≥ 0) operator on $\text{l}{}_{\mathrm{q}}{}^2$, the Hilbert space of square-summable complex-valued functions on q. Let A be p × q, i.e., let A be a linear operator from $\text{l}{}_{\mathrm{q}}{}^2$ to $\text{l}{}_{\mathrm{p}}{}^2$. Then exists a linear operator ǎ from (the Banach space) $\text{H}$_F^q to $\text{H}$_F^p on $\text{D}$(A) = {x:x ? $\text{H}$_F^q, $\text{A〈x, x〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ is p × q bounded on $\text{l}{}_{\mathrm{q}}{}^2$} such that y = ǎx satisfies y^j?σ(x) = {space spanned by the xⁱ}, 〈y, x〉 = A〈x, x〉 and $\text{〈y, y〉 = A〈x, x〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(A〈x, x〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}{}^1$. This is a generalization of our earlier [J. Multivariate Anal.4 (1974), 166–209; 6 (1976), 538–571] results for the case of a spectral measure concentrated on one point. We apply these tools to investigate q-variate wide-sense Markov processes.     

Phase reconstruction from magnitude of band-limited multidimensional signals

In this paper, the problem of phase reconstruction from magnitude of multidimensional band-limited functions is considered. It is shown that any irreducible band-limited function f(z₁…,z_n), z_i ? $\text{C}$, i=1, …, n, is uniquely determined from the magnitude of f(x₁…,x_n): | f(x₁…,x_n)|, x_i ? $\text{R}$, i=1,…, n, except for (1) linear shifts: ^{i(α1z1+…+α_n2n+β}), β, α_i?$\text{R}$, i=1,…, n; and (2) conjugation: $\text{f}{}^1\text{(z}{}_1{}^1\text{,…,z}{}_{\mathrm{n}}{}^1\text{)}$.     

Multipliers of Schatten classes

Let C_p be the class of all compact operators A on the Hilbert space l² for which $\text{∑¦λ}{}_{\mathrm{i}}\text{¦}{}^{\mathrm{p}}\text{< ∞}$, where {λ_i} is the eigen values of $\text{(A}{}^1\text{A)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$. The object of this paper is to prove some results concerning the Schur multipliers of C_p.     

A generalization of Lavoie's inequality concerning the sum of idempotent matrices

Let a complex pxn matrix A be partitioned as A′=(A′₁,A′₂,…,A′_k). Denote by ?(A), A′, and A^? respectively the rank of A, the transpose of A, and an inner inverse (or a g-inverse) of A. Let A⁽¹⁴⁾ be an inner inverse of A such that A⁽¹⁴⁾A is a Hermitian matrix. Let B=(A⁽¹⁴⁾₁,A⁽¹⁴⁾₂,…,A_k⁽¹⁴⁾) and $\text{ρ(A)=}\text{∑}\text{i=1}\text{k}\text{ρ(A}{}_{\mathrm{i}}\text{)}$.Then the product of nonzero eigenvalues of BA (or AB) cannot exceed one, and the product of nonzero eigenvalues of BA is equal to one if and only if either B=A⁽¹⁴⁾ or $\text{A}{}_{\mathrm{i>}}\text{A}{}_{\mathrm{<mtext>j</mtext>}}{}^1\text{=0}$ for all i ≠ j,i, j=1,2,…,k . The results of Lavoie (1980) and Styan (1981) are obtained as particular cases. A result is obtained for k=2 when the condition $\text{ρ(A)=}\text{∑}\text{i=1}\text{k}\text{ρ(A}{}_{\mathrm{i}}\text{)}$ is no longer true.     

Commuting self-adjoint partial differential operators and a group theoretic problem

In Rⁿ let Ω denote a Nikodym region (= a connected open set on which every distribution of finite Dirichlet integral is itself in $\text{L}{}^2\text{(Ω))}$. The existence of n commuting self-adjoint operators $\text{H}{}_1\text{,…, H}{}_{\mathrm{n}}\text{in}\text{L}{}^2\text{(Ω)}$ such that each H_j is a restriction of $\text{?i β}\text{βx}{}_{\mathrm{j}}$ (acting in the distribution sense) is shown to be equivalent to the existence of a set Λ ?Rⁿ such that the restrictions to Ω of the functions exp i ∑ λ_jx_j form a total orthogonal family in $\text{L}{}^2\text{(Ω)}$. If it is required, in addition, that the unitary groups generated by H₁,…, H_n act multiplicatively on $\text{L}{}^2\text{(Ω)}$, then this is shown to correspond to the requirement that Λ can be chosen as a subgroup of the additive group Rⁿ. The measurable sets Ω ?Rⁿ (of finite Lebesgue measure) for which there exists a subgroup Λ ?Rⁿ as stated are precisely those measurable sets which (after a correction by a null set) form a system of representatives for the quotient of Rⁿ by some subgroup Γ (essentially the dual of Λ).     

Duality in hypercomplex function theory

Let $\text{A}$ be the Clifford algebra constructed over a quadratic n-dimensional real vector space with orthogonal basis {e₁,…, e_n}, and e₀ be the identity of $\text{A}$. Furthermore, let M_k(Ω;$\text{A}$) be the set of $\text{A}$-valued functions defined in an open subset Ω of R^m+1 (1 ? m ? n) which satisfy D^kf = 0 in Ω, where D is the generalized Cauchy-Riemann operator $\text{D = ∑}{}_{\mathrm{i\; =\; 0}}{}^{\mathrm{m}}\text{e}{}_{\mathrm{i}}\text{(}\text{?}\text{?x}{}_{\mathrm{i}}\text{)}$ and k? N. The aim of this paper is to characterize the dual and bidual of M_k(Ω;$\text{A}$). It is proved that, if M_k(Ω;$\text{A}$) is provided with the topology of uniform compact convergence, then its strong dual is topologically isomorphic to an inductive limit space of Fréchet modules, which in its turn admits M_k(Ω;$\text{A}$) as its dual. In this way, classical results about the spaces of holomorphic functions and analytic functionals are generalized.