Generalized M-Matrices

Let ρ?Rⁿ be a proper cone. From the theory of M-matrices (see e.g. [1]) it is known that if there exist α > 0 and a matrix B: ρ→ρ such that A = B?αI, then the following conditions are equivalent: (i) ? A is ρ-monotone,(ii) A is ρ-seminegative, (iii) Re[Spectrum(A)]<0. In this paper we show that while the condition (e) e^tAρ?ρ ?t≥0 is more general than the structural assumption A = B?αI, conditions (i)-(iii) are nevertheless all equivalent to (iv) {x∈ρ: Ax∈ρ}={0} when (e) holds.     

A conjectured property of Hermitian pencils

If (A, B) is any pair of Hermitian matrices, the power of λ dividing det(λI?xA ?yB) will be given by the number of basic singular summands in the pair. Contrary to conjecture, this power can be greater than one even when the pair is unitarily irreducible.     

The computation of Kronecker's canonical form of a singular pencil

We develop stable algorithms for the computation of the Kronecker structure of an arbitrary pencil. This problem can be viewed as a generalization of the well-known eigenvalue problem of pencils of the type λI?A. We first show that the elementary divisors (λ ? α)ⁱ of a regular pencil λB?A can be retrieved with a deflation algorithm acting on the expansion (λ ? α)B ? (A ? αB). This method is a straightforward generalization of Kublanovskaya's algorithm for the determination of the Jordan structure of a constant matrix. We also show how to use this method to determine the structure of the infinite elementary divisors of λB?A. In the case of singular pencils, the occurrence of Kronecker indices—containing the singularity of the pencil—somewhat complicates the problem. Yet our algorithm retrieves these indices with no additional effort, when determining the elementary divisors of the pencil. The present ideas can also be used to separate from an arbitrary pencil a smaller regular pencil containing only the finite elementary divisors of the original one. This is shown to be an effective tool when used together with the QZ algorithm.     

A note on a new matrix decomposition

It is shown that if det A=±1, then A=±^q_i=1B_i, where B_i² = I. This decomposition is used to find the Jacobian of the linear matrix transformation: Y=AX.     

ℓ-group homomorphisms between reduced archimedean f-rings

Let A and B be reduced archimedean f-rings, A with identity e; let $A\,\mathop \to \limits^\gamma\,BLet A and B be reduced archimedean f-rings, A with identity e; let A \mathop ? ^g BA\,\mathop \to \limits^\gamma\,B be an ℓ-group homomorphism, and set w = γ (e). We show (with some vagaries of phrasing here) (1) γ = w·ρ for a canonical ℓ-ring homomorphism A \mathop ? ^r B (w)A\,\mathop \to \limits^\rho\,B (w), where B (w) is an extension of B in which w is a von Neumann regular element, and (2) for X _A ,X _B canonical representation spaces for A, B, γ is realized via composition with a unique partially defined continuous function from X _B to X _A .     

C0-semigroups on a locally convex space

A necessary and sufficient condition that a densely defined linear operator A in a sequentially complete locally convex space X be the infinitesimal generator of a quasi-equicontinuous C₀-semigroup on X is that there exist a real number β ? 0 such that, for each λ > β, the resolvent (λI ? A)^?1 exists and the family {(λ ? β)^k(λI ? A)^?k; λ > β, k = 0, 1, 2,…} is equicontinuous. In this case all resolvents (λI ? A)^?1, λ > β, of the given operator A and all exponentials exp(tA), t ? 0, of the operator A belong to a Banach algebra $\text{B}{}_{\mathrm{г}}\text{(X)}$ which is a subspace of the space L(X) of all continuous linear operators on X, and, for each t ? 0 and for each x?X, one has lim_{k → z} (I ? k^?1tA)^?kx = exp(tA) x. A perturbation theorem for the infinitesimal generator of a quasi-equicontinuous C₀-semigroup by an operator which is an element of $\text{B}{}_{\mathrm{г}}\text{(X)}$ is obtained.     

On simultaneous diagonalization of one Hermitian and one symmetric form

It is remarked that if A, B ? M_n(C), A = A^t, and B? = B^t, B positive definite, there exists a nonsingular matrix U such that (1) ū^tBU = I and (2) U^tAU is a diagonal matrix with nonnegative entries. Some related actions of the real orthogonal group and equations involving the unitary group are studied.     

Sequence entropy and mixing

The relationship between sequence entropy and mixing is examined. Let T be an automorphism of a Lebesgue space X, $\text{L}$₀ denote the set of all partitions of X possessing finite entropy, and $\text{S}$ denote the set of all increasing sequences of positive integers. It is shown that: (1) T is mixing /a2 sup_{A ? B}h_A(T, α) = H(α) for all B∈I and α∈Z₀. (2) T is weakly mixing /a2 sup_Ah_A(T, α) = H(α) for all α∈Z₀. (3) If T is partially mixing with constant $\text{c (1 ?}\text{1}\text{e}\text{< c < 1)}$, then sup_{A ? B}h_A(T, α) > cH(α) for all B∈I and nontrivial α∈Z₀. (4) If sup_{A ? B}h_A(T, α) > 0 for all B∈I and nontrivial α∈Z₀, then T is weakly mixing.     

MAPS PRESERVING STRONG SKEW LIE PRODUCT ON FACTOR VON NEUMANN ALGEBRAS

Let A be a factor von Neumann algebra and Φ be a nonlinear surjective map from A onto itself.We prove that,if Φ satisfies that Φ(A)Φ(B) - Φ(B)Φ(A)* =AB - BA* for all A,B ∈ A,then there exist a linear b...     

A determinantal inequality involving the Moore-Penrose inverse

Let A be a rectangular matrix of complex numbers whose rows are partitioned into r arbitrary blocks:

The Moore-Penrose inverses of each of these blocks are used to form the matrix B = (A₁⁺,…, A_r⁺). It is shown that 0 ? det (AB) ? 1. This is a generalized version of Hadamard's inequality.     

Strongly e-movable convergence and spaces of ANR's

Let X be a metric space and let ANR(X) denote the hyperspace of all compact ANR's in X. This paper introduces a notion of a strongly e-movable convergence for sequences in ANR(X) and proves several characterizations of strongly e-movable convergence. For a (complete) separable metric space X we show that ANR(X) with the topology induced by strongly e-movable convergence can be metrized as a (complete) separable metric space. Moreover, if X is a finite-dimensional compactum, then strongly e-movable convergence induces on ANR(X) the same topology as that induced by Borsuk's homotopy metric.For a separable Q-manifold M, ANR(M) is locally arcwise connected and A, B ? ANR(M) can be joined by an arc in ANR(M) iff there is a simple homotopy equivalence ?: A → B homotopic to the inclusion of A into M.     

On the evaluation of the Eigenvalues of a banded toeplitz block matrix

Let A = (a_ij) be an n × n Toeplitz matrix with bandwidth k + 1, K = r + s, that is, a_ij = a_j−i, i, J = 1,… ,n, a_i = 0 if i > s and if i < -r. We compute p(λ)= det(A - λI), as well as p(λ)/p′(λ), where p′(λ) is the first derivative of p(λ), by using O(k log k log n) arithmetic operations. Moreover, if a_i are m × m matrices, so that A is a banded Toeplitz block matrix, then we compute p(λ), as well as p(λ)/p′(λ), by using O(m³k(log² k + log n) + m²k log k log n) arithmetic operations. The algorithms can be extended to the computation of det(A − λB) and of its first derivative, where both A and B are banded Toeplitz matrices. The algorithms may be used as a basis for iterative solution of the eigenvalue problem for the matrix A and of the generalized eigenvalue problem for A and B.     

On the commuting extensions of subnormal operators

We give some results concerning the following problem: Given a linear bounded operatorA which is subnormal on a Hilbert spaceH, andB its minimal normal extension on a Hilbert spaceK ～H, when can a quasi-normal operatorT commuting withA be extended to an operatorT ^e onK such thatT ^e commutes withB andT ^e is quasi-normal onK?     

On the Cauchy problem for some abstract nonlinear differential equations

In the present paper, we study the Cauchy problem in a Banach spaceE for an abstract nonlinear differential equation of form $$\frac{{d^2 u}}{{dt^2 }} = - A\frac{{du}}{{dt}} + B(t)u + f(t,W)$$ whereW = (A ₁(t)u,A ₂(t)u,?,A _?(t)u), (A _i (t),i = 1, 2, ?,?), (B(t),t ∈I = [0,b]) are families of closed operators defined on dense sets inE intoE, f is a given abstract nonlinear function onI ×E ^? intoE and ?A is a closed linear operator defined on dense set inE intoE, which generates a semi-group. Further, the existence and uniqueness of the solution of the considered Cauchy problem is studied for a wide class of the families (A _i(t),i = 1, 2, ?,?), (B(t),t ∈I). An application and some properties are also given for the theory of partial diferential equations.     

A duality between locally compact groups and certain Banach algebras

We make precise the following statements: B(G), the Fourier-Stieltjes algebra of locally compact group G, is a dual of G and vice versa. Similarly, A(G), the Fourier algebra of G, is a dual of G and vice versa. We define an abstract Fourier (respectively, Fourier-Stieltjes) algebra; we define the dual group of such a Fourier (respectively, Fourier-Stieltjes) algebra; and we prove the analog of the Pontriagin duality theorem in this context. The key idea in the proof is the characterization of translations of B(G) as precisely those isometric automorphisms Φ of B(G) which satisfy ∥ p ? e^iθΦp ∥² + ∥ p + e^iθΦp ∥² = 4 for all θ ∈ $\text{R}$ and all pure positive definite functions p with norm one. One particularly interesting technical result appears, namely, given x₁, x₂?G, neither of which is the identity e of G, then there exists a continuous, irreducible unitary representation π of G (which may be chosen from the reduced dual of G) such that π(x₁) ≠ π(e) and π(x₂) ≠ π(e). We also note that the group of isometric automorphisms of B(G) (or A(G)) contains as a (“large”) ^.closed, normal subgroup the topological version of Burnside's “holomorph of G.”     

Symplectic groups and the Klein-Gordon field

In this paper a Cohen factorization theorem x = a^t · x_t (t > 0) is proved for a Banach algebra A with a bounded approximate identity, where t ? a^t is a continuous one-parameter semigroup in A. This theorem is used to show that a separable Banach algebra B has a bounded approximate identity bounded by 1 if and only if there is a homomorphism θ from L¹($\text{R}$⁺) into B such that ∥ θ ∥ = 1 and θ(L¹($\text{R}$⁺)). B = B = B · θ(L¹($\text{R}$⁺)). Another corollary is that a separable Banach algebra with bounded approximate identity has a commutative bounded approximate identity, which is bounded by 1 in an equivalent algebra norm.     

Involutory functions and Moore-Penrose inverses of matrices in an arbitrary field

Let F be a field, and M be the set of all matrices over F. A function ? from M into M, which we write ?(A) = A^s for A∈M, is involutory if (1) (AB)^s = B^sA^s for all A, B in M whenever the product AB is defined, and (2) (A^s)^s = A for all A∈M. If ? is an involutory function on M, then A^s is n×m if A is m×n; furthermore, Rank A = Rank A^s, the restriction of ? to F is an involutory automorphism of F, and (aA + bB)^s = a^sA^s + b^sB^s for all m×n matrices A and B and all scalars a and b. For an A∈M, an Ã∈M is called a Moore-Penrose inverse of A relative to ? if (i) AÃA = A, ÃAÃ = Ã and (ii) (AÃ)^s = AÃ, (ÃA)^s = ÃA. A necessary and sufficient condition for A to have a Moore-Penrose inverse relative to ? is that Rank A = Rank AA^s = Rank A^sA. Furthermore, if an involutory function ? preserves circulant matrices, then the Moore-Penrose inverse of any circulant matrix relative to ? is also circulant, if it exists.     

Global inverse spectral problems for rational matrix functions

Given rational matrix functions ψ₁(λ) = I_m + C₁(λI_n1 − A₁)⁻¹B₁ and ψ₂(λ) = I_m + C₂(λI_n2 − A₂)⁻¹B₂ which are analytic and invertible on the unit circle, we characterize in terms of the operators A₁,B₁,C₁,A₂,B₂,C₂ when there exists a single rational matrix function W(λ) = I_m + C(λI_n − A)⁻¹B such that WH²_m^⊥ = ψ ₁H²_m^⊥and WH²_m = ψ₂H²_m. When this is the case, we give explicit formulae for A,B,C in terms of A₁,B₁,C₁,A₂,B₂,C₂. Applications include Wiener-Hopf factorization, J- inner-outer factorization, and coprime factorization. The results on J-inner-outer factorization have application to a model reduction problem for discrete time linear systems.     

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.     

On stabilizability of evolution systems of partial differential equations on ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>×[0,+∞) by time-delayed feedback controlsby time-delayed feedback controls

In this work we obtain sufficient conditions for stabilizability by time-delayed feedback controls for the system

$\frac{{\partial w\left( {x,t} \right)}}{{\partial t}} = A(D_x )w(x,t) - A(D_x )u(x,t), x \in \mathbb{R}^n , t > h, $

where D _x =(-i?/?x ₁,...-i?/?x _n ), A(σ) and B(σ) are polynomial matrices (m×m), det B(σ)≡0 on ? ⁿ , w is an unknown function, u(·,t)=P(D _x )w(·,t?h) is a control, h>0. Here P is an infinite differentiable matrix (m×m), and the norm of each of its derivatives does not exceed Γ(1+|σ|²)^γ for some Γ, γ∈? depending on the order of this derivative. Necessary conditions for stabilizability of this system are also obtained. In particular, we study the stabilizability problem for the systems corresponding to the telegraph equation, the wave equation, the heat equation, the Schrödinger equation and another model equation. To obtain these results we use the Fourier transform method, the Lojasiewicz inequality and the Tarski—Seidenberg theorem and its corollaries. To choose an appropriate P and stabilize this system, we also prove some estimates of the real parts of the zeros of the quasipolynomial det {Iλ-A(σ)+B(σ)P(σ)e ^-hλ.