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.     

Cesàro asymptotics for the orders of SLk(Zn) and GLk(Zn) as n→∞

Given an integer k>0, our main result states that the sequence of orders of the groups $\text{SL}{}_{\mathrm{k}}\text{(}\text{Z}{}_{\mathrm{n}}\text{)}$ (respectively, of the groups $\text{GL}{}_{\mathrm{k}}\text{(}\text{Z}{}_{\mathrm{n}}\text{)}$) is Cesàro equivalent as n→∞ to the sequence C₁(k)n^k2?1 (respectively, C₂(k)n^k2), where the coefficients C₁(k) and C₂(k) depend only on k; we give explicit formulas for C₁(k) and C₂(k). This result generalizes the theorem (which was first published by I. Schoenberg) that says that the Euler function ?(n) is Cesàro equivalent to $\text{n}\text{6}\text{π}{}^2$. We present some experimental facts related to the main result. To cite this article: A.G. Gorinov, S.V. Shadchin, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Injectivity and operator spaces

Let B(H) be the bounded operators on a Hilbert space H. A linear subspace R ? B(H) is said to be an operator system if 1 ?R and R is self-adjoint. Consider the category $\text{b}$ of operator systems and completely positive linear maps. R ∈ $\text{C}$ is said to be injective if given A ? B, A, B ∈ $\text{C}$, each map A → R extends to B. Then each injective operator system is isomorphic to a conditionally complete C¹-algebra. Injective von Neumann algebras R are characterized by any one of the following: (1) a relative interpolation property, (2) a finite “projectivity” property, (3) letting M_m = B(C^m), each map R → N ? M_m has approximate factorizations R → M_n → N, (4) letting K be the orthogonal complement of an operator system N ? M_m, each map $\text{M}{}_{\mathrm{m}}\text{K}\text{→ R}$ has approximate factorizations $\text{M}{}_{\mathrm{m}}\text{K}\text{→ M}{}_{\mathrm{n}}\text{→ R}$. Analogous characterizations are found for certain classes of C¹-algebras.     

On tensor products of certain group C1-algebras

If A and B are C¹-algebras there is, in general, a multiplicity of C¹-norms on their algebraic tensor product A ⊙ B, including maximal and minimal norms ν and α, respectively. A is said to be nuclear if α and ν coincide, for arbitrary B. The earliest example, due to Takesaki [11], of a nonnuclear C¹-algebra was $\text{C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{)}$, the C¹-algebra generated by the left regular representation of the free group on two generators F₂. It is shown here that W¹-algebras, with the exception of certain finite type I's, are nonnuclear.If $\text{C}{}^1\text{(F}{}_2\text{)}$ is the group C¹-algebra of F₂, there is a canonical homomorphism λ_l of $\text{C}{}^1\text{(F}{}_2\text{)}$ onto $\text{C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{)}$. The principal result of this paper is that there is a norm ζ on $\text{C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{) ⊙ C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{)}$, distinct from α, relative to which the homomorphism $\text{λ ⊙ λ}{}_{\mathrm{l}}\text{: C}{}^1\text{(F}{}_2\text{) ⊙ C}{}^1\text{(F}{}_2\text{) → C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{) ⊙ C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{)}$ is bounded ($\text{C}{}^1\text{(F}{}_2\text{) ⊙ C}{}^1\text{(F}{}_2\text{)}$ being endowed with the norm α). Thus quotients do not, in general, respect the norm α; a consequence of this is that the set of ideals of the α-tensor product of C¹-algebras A and B may properly contain the set of product ideals {$\text{I ? B + A ? J: I ? A, J ? B}$}.Let A and B be C¹-algebras. If A or B is a W¹-algebra there are on A ⊙ B certain C¹-norms, defined recently by Effros and Lance [3], the definitions of which take account of normality. In the final section of the paper it is shown by example that these norms, with α and ν, can be mutually distinct.     

Parallel calculation of a linear mapping on a computer network

We are interested in the parallel computation of a linear mapping of n real variables by a network of computers with restricted means of communication between them and without any common memory. Let $\text{M}{}_{\mathrm{n\times n}}\text{(}\text{R}\text{)}$ denote the algebra of n×n real matrices, and let G be the graph associated with a binary, reflexive and symmetric relation R over {1,2, …,n}. We define

$\text{A}{}_{\mathrm{R}}\text{= {A?M}{}_{\mathrm{n\times n}}\text{(}\text{R}\text{):a}{}_{\mathrm{ij}}\text{≠ 0}\text{implies}\text{iRj}}$

A matrix $\text{M∈M}{}_{\mathrm{n\times n}}\text{(}\text{R}\text{)}$ is said to be realizable on G if it can be expressed as a product of elements of A_R. Therefore, every matrix of $\text{M}{}_{\mathrm{n\times n}}\text{(}\text{R}\text{)}$ is realizable on G if and only if A_R generates $\text{M}{}_{\mathrm{n\times n}}\text{(}\text{R}\text{)}$. We show that A_R generates $\text{M}{}_{\mathrm{n\times n}}\text{(}\text{R}\text{)}$ if and only if G is connected.     

Generalized Schur complements

Let A be an n×n complex matrix. For a suitable subspace $\text{M}$ of Cⁿ the Schur compression A _$\text{M}$ and the (generalized) Schur complement A_/$\text{M}$ are defined. If A is written in the form

$\text{A=}\text{B}\text{C}\text{S}\text{T}$

according to the decomposition $\text{C}{}^{\mathrm{n}}\text{=}\text{M}\text{⊕}\text{M}{}^{\mathrm{\perp }}$ and if B is invertible, then

$\text{A}{}_{\mathrm{<mtext>M</mtext>}}\text{=}\text{B}\text{C}\text{S}\text{SB}{}^{\mathrm{?1}}\text{C}$

and

$\text{A}{}_{\mathrm{<mtext>/</mtext><mtext>M</mtext>}}\text{=}\text{0}\text{0}\text{0}\text{T?SB}{}^{\mathrm{?1}}\text{C}\text{·}$

The commutativity rule for Schur complements is proved:

$\text{(A}{}_{\mathrm{<mtext>/</mtext><mtext>M</mtext>}}\text{)}{}_{\mathrm{<mtext>/</mtext><mtext>N</mtext>}}\text{=(A)}{}_{\mathrm{<mtext>/</mtext><mtext>N</mtext>}}\text{)}{}_{\mathrm{<mtext>/</mtext><mtext>M</mtext>}}\text{·}$

This unifies Crabtree and Haynsworth's quotient formula for (classical) Schur complements and Anderson's commutativity rule for shorted operators. Further, the absorption rule for Schur compressions is proved:

$\text{(A}{}_{\mathrm{<mtext>/</mtext><mtext>M</mtext>}}\text{)}{}_{\mathrm{<mtext>N</mtext>}}\text{=(A}{}_{\mathrm{<mtext>N</mtext>}}\text{)}{}_{\mathrm{<mtext>M</mtext>}}\text{=A}{}_{\mathrm{<mtext>M</mtext>}}\text{whenever}\text{M}\text{?}\text{N}$

.     

Subproper splitting for rectangular matrices

In this paper iterative schemes for approximating a solution to a rectangular but consistent linear system Ax = b are studied. Let A?C^{m × n}_r. The splitting A = M ? N is called subproper if R(A) ? R(M) and $\text{R(A}{}^1\text{) ?R(M}{}^1\text{)}$. Consider the iteration $\text{x}{}_{\mathrm{i}}\text{= M}{}^2\text{Nx}{}_{\mathrm{i}}{}_{\mathrm{?1}}\text{+ M}{}^2\text{b}$. We characterize the convergence of this scheme to a solution of the linear system. When A?R^m×ⁿ_r, monotonicity and the concept of subproper regular splitting are used to determine a necessary and a sufficient condition for the scheme to converge to a solution.     

Multiplicativity of lp norms for matrices. II

For 1 ? p ? ∞, let

$\text{|A|}{}_{\mathrm{p}}\text{=}\text{Σ}\text{i=1}\text{m}\text{Σ}\text{j=1}\text{n}\text{, |α}{}_{\mathrm{ij}}\text{|}{}^{\mathrm{p}}{}^{\mathrm{<mtext>1</mtext><mtext>p</mtext>}}$

, be the l_p norm of an m × n complex A = (α_ij) ?C_{m × n}. The main purpose of this paper is to find, for any p, q ? 1, the best (smallest) possible constants τ(m, k, n, p, q) and σ(m, k, n, p, q) for which inequalities of the form

$\text{|AB|}{}_{\mathrm{p}}\text{? τ(m, k, n, p, q) |A|}{}_{\mathrm{p}}\text{|B|}{}_{\mathrm{q}}\text{, |AB|}{}_{\mathrm{p}}\text{? σ (m, k, n, p, q)|A|}{}_{\mathrm{q}}\text{|B|}{}_{\mathrm{p}}$

hold for all A?C_{m × k}, B?C_{k × n}. This leads to upper bounds for inner products on C^k and for ordinary l_p operator norms on C_{m × n}.     

On permutations,convex hulls,and normal operators

A spectral characterization is obtained for those normal operators which belong to the convex hull of the unitary orbit of a given normal operator on a finite-dimensional space. This is used to prove the following: if A and B are normal operators on an n-dimensional complex Hilbert space H with eigenvalues given by α₁,…,α_n and β₁,…, β_n respectively, and if A ? B is also normal, then $\text{6A ? B6 ?}\text{max}\text{σ ? S}{}_{\mathrm{n}}\text{6}\text{diag}\text{(α}{}_{\mathrm{k}}\text{?β}{}_{\mathrm{\sigma (k)}}\text{)6}$ for any unitarily invariant norm on L(H).     

Certain isometries of rectangular complex matrices

If s₁(A) ? ? ? s_m(A) are the singular values of A ? M_m,n(C), and if 1 ?k ?m ? and p ? 1, then

$\text{φ}{}_{\mathrm{p,k}}\text{(A) = (}\text{∑}\text{i=1}\text{k}\text{s}{}_{\mathrm{i}}{}^{\mathrm{p}}\text{(A)}{}^{\mathrm{<mtext>1</mtext><mtext>p</mtext>}}$

is a unitarily invariant norm. In this paper a complete determination of the extreme points on the corresponding unit spheres is accomplished in all cases, enabling the isometries with respect to Φ_p,_k to be determined in the case p = 1. This removes the restriction m = n in an earlier paper of the author and Marcus.     

Asymptotic results for partitions (II) and a conjecture of Bateman and Erdös

Asymptotic results are obtained for p_A^(k)(n), the kth difference of the function p_A(n) which is the number of partitions of n into integers from A. Under certain restrictions on A it is shown that

$\text{P}{}_{\mathrm{A}}{}^{\mathrm{(k+1)}}\text{(n)}\text{P}{}_{\mathrm{A}}{}^{\mathrm{(k)}}\text{(n)}\text{= O(n}{}^{\mathrm{?1/2}}\text{) (n→ ∫)}$

thereby verifying for these A a conjecture of Bateman and Erdös.     

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.     

Spaces of matrices of equal rank

Let M_m,n(F) denote the space of all mXn matrices over the algebraically closed field F. A subspace of M_m,n(F), all of whose nonzero elements have rank k, is said to be essentially decomposable if there exist nonsingular mXn matrices U and V respectively such that for any element A, UAV has the form

$\text{UAV=}\text{A}{}_1\text{A}{}_2\text{A}{}_3\text{0}$

where A₁ is iX(k–i) for some i?k. Theorem: If $\text{K}$ is a space of rank k matrices, then either $\text{K}$ is essentially decomposable or dim $\text{K}$?k+1. An example shows that the above bound on non-essentially-decomposable spaces of rank k matrices is sharp whenever n?2k–1.     

On essential self-adjointness for Schrödinger operators with wildly oscillating potentials

A sufficient condition is given for the operator T₀: C^∞₀(R^m) → L²(R^m) given by

$\text{T}{}_0\text{∑}\text{K?1}\text{M}\text{(}\text{i?}\text{?x}{}_1\text{+b}{}_1\text{)a}{}_{\mathrm{1K}}\text{(}\text{i?}\text{?N}{}_{\mathrm{k}}\text{+b}{}_{\mathrm{k}}\text{)+q}$

to be essentially self-adjoint. This condition is sufficiently general to admit certain potentials q having unbounded oscillations in a neighborhood of ∞.     

Third order efficiency of conditional tests for exponential families

Let P_η, η = (θ, γ) ∈ Θ × Γ ? $\text{R}$ × $\text{R}$^k, be a (k + 1)-dimensional exponential family. Let $\text{?}{}_{\mathrm{n}}{}^1$, n ∈ $\text{N}$, be an optimal similar test for the hypothesis {P_(θ,γ)ⁿ: γ ∈ Γ} (θ ∈ Θ fixed) against alternatives P_(θ1,γ1)ⁿ, θ₁ > θ, γ₁ ∈ Γ. It is shown that (?_n¹)_{n∈$\text{N}$} is third order efficient in the class of all test-sequences that are asymptotically similar of level α + o(n^?1) (locally uniformly in the nuisance parameter γ).     

The minimal generating space of a polynomial

Given a polynomial $\text{P(X}{}_1\text{,…,X}{}_{\mathrm{N}}\text{)∈}\text{R}\text{[}\text{X}\text{]}$, we calculate a subspace G_p of the linear space 〈X〉 generated by the indeterminates which is minimal with respect to the property $\text{P∈}\text{R}\text{[}\text{G}{}_{\mathrm{p}}\text{]}$ (the algebra generated by G_p, and prove its uniqueness. Furthermore, we use this result to characterize the pairs (P,Q) of polynomials P(X₁,…,X_n) and Q(X₁,…,X_n) for which there exists an isomorphism T:〈X〉 →〈X〉 that “separates P from Q,” i.e., such that for some k(1<k<n) we can write P and Q as $\text{P}{}^1\text{(Y}{}_1\text{,…,Y}{}_{\mathrm{k}}\text{)}$ and $\text{Q}{}^1\text{(Y}{}_{\mathrm{<mtext>k+</mtext><mtext>1</mtext>}}\text{,…,Y}{}_{\mathrm{n}}\text{)}$ respectively, where $\text{Y}\text{=T}\text{X}$.     

On matrix functions which commute with their derivative

We improve several results published from 1950 up to 1982 on matrix functions commuting with their derivative, and establish two results of general interest. The first one gives a condition for a finite-dimensional vector subspace E(t) of a normed space not to depend on t, when t varies in a normed space. The second one asserts that if A is a matrix function, defined on a set ?, of the form A(t)= U diag(B₁(t),…,B_p(t)) U^-1, t ∈ ?, and if each matrix function B_k has the polynomial form

$\text{B}{}_{\mathrm{k}}\text{(t)=}\text{∑}\text{i=0}\text{α}{}_{\mathrm{k}}\text{f}{}_{\mathrm{ki}}\text{(t)C}{}_{\mathrm{k}}{}^{\mathrm{i}}\text{, t∈ ?, k∈{1,…,p}}$

then A itself has the polynomial form

$\text{A(t)=}\text{∑}\text{i=0}\text{d?1}\text{f}{}_{\mathrm{i}}\text{(t)C}{}^{\mathrm{i}}\text{,t∈?}$

, where

$\text{d=}\text{∑}\text{k=1}\text{p}\text{d}{}_{\mathrm{k}}$

, d_k being the degree of the minimal polynomial of the matrix C_k, for every k ∈ {1,…,p}.     

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.