Linear preservers of matrices of rank-2

Let T be a linear operator on the space of all m×n matrices over any field. we prove that if T maps rank-2 matrices to rank-2 matrices then there exist nonsingular matrices U and V such that either T(X)=UXV for all matrices X, or m=n and T(X)=UX^tV for all matrices X where X^t denotes the transpose of X.     

Generating orthogonal matrix polynomials satisfying second order differential equations from a trio of triangular matrices

The method developed in [A.J. Durán, F.A. Grünbaum, Orthogonal matrix polynomials satisfying second order differential equations, Int. Math. Res. Not. 10 (2004) 461–484] led us to consider matrix polynomials that are orthogonal with respect to weight matrices W(t) of the form , , and (1−t)^α(1+t)^βT(t)T^*(t), with T satisfying T^′=(2Bt+A)T, T(0)=I, T^′=(A+B/t)T, T(1)=I, and T^′(t)=(−A/(1−t)+B/(1+t))T, T(0)=I, respectively. Here A and B are in general two non-commuting matrices. We are interested in sequences of orthogonal polynomials (P_n)_n which also satisfy a second order differential equation with differential coefficients that are matrix polynomials F₂, F₁ and F₀ (independent of n) of degrees not bigger than 2, 1 and 0 respectively. To proceed further and find situations where these second order differential equations hold, we only dealt with the case when one of the matrices A or B vanishes.The purpose of this paper is to show a method which allows us to deal with the case when A, B and F₀ are simultaneously triangularizable (but without making any commutativity assumption).     

On nonautonomous linear systems of differential and difference equations with -symmetric coefficient matrices

Let denote the set of continuous n×n matrices on an interval . We say that is a nontrivial k-involution if where ζ=e^-2πi/k, d₀+d₁++d_k-1=n, and with . We say that is R-symmetric if R(t)A(t)R^-1(t)=A(t), , and we show that if A is R-symmetric then solving x^′=A(t)x or x^′=A(t)x+f(t) reduces to solving k independent d_ℓ×d_ℓ systems, 0ℓk-1. We consider the asymptotic behavior of the solutions in the case where . Finally, we sketch analogous results for linear systems of difference equations.     

Linear transformations on symmetric matrices that preserve commutativity

It is shown that if a nonsingular linear transformation T on the space of n-square real symmetric matrices preserves the commutativity, where n ?3, then T(A) = λQAQ^t + Q(A)I_n for all symmetric matricesA, for some scalar λ, orthogonal matrix Q, and linear functional Q.     

Wellposedness of abstract Cauchy problems for second order differential equations

This paper concerns the abstract Cauchy problem (ACP) for an evolution equation of second order in time. LetA be a closed linear operator with domainD(A) dense in a Banach spaceX. We first characterize the exponential wellposedness of ACP onD(A ^k+1),k teN. Next let {C(t);t teR} be a family of generalized solution operators, on [D(A ^k)] toX, associated with an exponentially wellposed ACP onD(A ^k+1). Then we define a new family {T(t); Ret>0} by the abstract Weierstrass formula. We show that {T(t)} forms a holomorphic semigroup of class (H _k) onX. Research of the second-named author was partially supported by Grant-in-Aid for Scientific Research (No. 63540139), Ministry of Education, Science and Culture.     

Parameter-dependent operator equations of the Riccati type with application to transport theory

We obtain an existence result for global solutions to initial-value problems for Riccati equations of the form R′(t) + TR(t) + R(t)T = T^ρ A(t)T^1?ρ + T^ρ B(t)T^1?ρ R(t) + R(t)T^ρC(t) T^1?ρ + R(t)T^ρD(t)T^1?ρ R(t), R(0)=R₀, where 0 ? ρ ? 1 and where the functions R and A through D take on values in the cone of non-negative bounded linear operators on L¹ (0, W; μ). T is an unbounded multiplication operator. This problem is of particular interest in case ρ = 1 since it arisess in the theories of particle transport and radiative transfer in a slab. However, in this case there are some serious difficulties associated with this equation, which lead us to define a solution for the case ρ = 1 as the limit of solutions for the cases 0 < ρ < 1.     

Bijective linear maps on semimodules spanned by Boolean matrices of fixed rank

Let M_m,n(B) be the semimodule of all m×n Boolean matrices where B is the Boolean algebra with two elements. Let k be a positive integer such that 2?k?min(m,n). Let B(m,n,k) denote the subsemimodule of M_m,n(B) spanned by the set of all rank k matrices. We show that if T is a bijective linear mapping on B(m,n,k), then there exist permutation matrices P and Q such that T(A)=PAQ for all A∈B(m,n,k) or m=n and T(A)=PA^tQ for all A∈B(m,n,k). This result follows from a more general theorem we prove concerning the structure of linear mappings on B(m,n,k) that preserve both the weight of each matrix and rank one matrices of weight k². Here the weight of a Boolean matrix is the number of its nonzero entries.     

C-Cosine Functions and the Abstract Cauchy Problem, II

For a bounded linear injectionCon a Banach spaceXand a closed linear operatorA : D(A) X → Xwhich commutes withCwe prove that (1) the abstract Cauchy problem,u″(t) = Au(t),t R,u(0) = Cx,u′(0) = Cy, has a unique strong solution for everyx,y D(A) if and only if (2)A₁ = AD(A²) generates aC₁-cosine function onX₁(D(A) with the graph norm), if (and only if, in caseAhas nonempty resolvent set) (3)Agenerates aC-cosine function onX. HereC₁ = CX₁. Under the assumption thatAis densely defined andC⁻¹AC = A, statement (3) is also equivalent to each of the following statements: (4) the problemv″(t) = Av(t) + C(x + ty) + ∫^t₀ Cg(r) dr,t R,v(0) = v′(0) = 0, has a unique strong solution for everyg L¹_locandx, y X; (5) the problemw″(t) = Aw(t) + Cg(t),t R,w(0) = Cx,w′(0) = Cy, has a unique weak solution for everyg L¹_locandx, y X. Finally, as an application, it is shown that for any bounded operatorBwhich commutes withCand has range contained in the range ofC,A + Bis also a generator.     

Randomly fractionally integrated processes

Philippe et al. [9], [10] introduced two distinct time-varying mutually invertible fractionally integrated filters A(d), B(d) depending on an arbitrary sequence d = (d _t ) _t∈ℤ of real numbers; if the parameter sequence is constant d _t ≡ d, then both filters A(d) and B(d) reduce to the usual fractional integration operator (1 − L)^−d . They also studied partial sums limits of filtered white noise nonstationary processes A(d)ε _t and B(d)ε _t for certain classes of deterministic sequences d. The present paper discusses the randomly fractionally integrated stationary processes X _t ^A = A(d)ε _t and X _t ^B = B(d)ε _t by assuming that d = (d _t , t ∈ ℤ) is a random iid sequence, independent of the noise (ε _t ). In the case where the mean , we show that large sample properties of X ^A and X ^B are similar to FARIMA(0, , 0) process; in particular, their partial sums converge to a fractional Brownian motion with parameter . The most technical part of the paper is the study and characterization of limit distributions of partial sums for nonlinear functions h(X _t ^A ) of a randomly fractionally integrated process X _t ^A with Gaussian noise. We prove that the limit distribution of those sums is determined by a conditional Hermite rank of h. For the special case of a constant deterministic sequence d _t , this reduces to the standard Hermite rank used in Dobrushin and Major [2]. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 1, pp. 3–28, January–March, 2007.     

Stability of <Emphasis Type="Italic">C</Emphasis>-regularized Semigroups

Let T = (T(t))t≥0 be a bounded C-regularized semigroup generated by A on a Banach space X and R(C) be dense in X. We show that if there is a dense subspace Y of X such that for every x ∈ Y, σu(A, Cx), the set of all points λ ∈ iR to which (λ - A)^-1 Cx can not be extended holomorphically, is at most countable and σr(A) N iR = Ф, then T is stable. A stability result for the case of R(C) being non-dense is also given. Our results generalize the work on the stability of strongly continuous senfigroups.     

Additive Maps Preserving Similarity of Operators on Banach Spaces

Let X be an infinite-dimensional complex Banach space and denote by B（X） the algebra of all bounded linear operators acting on X. It is shown that a surjective additive map Φ from B（X） onto itself preserves similarity in both directions if and only if there exist a scalar c, a bounded invertible linear or conjugate linear operator A and a similarity invariant additive functional ψ on B（X） such that either Φ（T） = cATA^-1 ＋ ψ（T）I for all T, or Φ（T） = cAT＊A^-1 ＋ ψ（T）I for all T. In the case where X has infinite multiplicity, in particular, when X is an infinite-dimensional Hilbert space, the above similarity invariant additive functional ψ is always zero.     

Asymptotic Behaviour of C0–Semigroups with Bounded Local Resolvents

Let {T(t)}_t≥0 be a C₀–semigroup on a Banach space X with generator A, and let H^∞_T be the space of all x ∈ X such that the local resolvent λ ↦ R(λ, A)x has a bounded holomorphic extension to the right half–plane. For the class of integrable functions ϕ on [0, ∞) whose Fourier transforms are integrable, we construct a functional calculus ϕ ↦ T_ϕ, as operators on H^∞_T. Weshow that each orbit T(·)T_ϕx is bounded and uniformly continuous, and T(t)T_ϕx → 0 weakly as t → ∞, and we give a new proof that ∥T(t)R(μ, A)x∥ = O(t). We also show that ∥T(t)T_ϕx∥ → 0 when T is sun –reflexive, and that ∥T(t)R(μ, A)x∥ = O(ln t) when T is a positive semigroup on a normal ordered space X and x is a positive vector in H^∞_T.     

The smooth Banach submanifold B*(E, F) in B(E, F)

Let E,F be two Banach spaces,B(E,F),B+(E,F),Φ(E,F),SΦ(E,F) and R(E,F) be bounded linear,double splitting,Fredholm,semi-Frdholm and finite rank operators from E into F,respectively. Let Σ be any one of the following sets:{T ∈Φ(E,F):Index T=constant and dim N(T)=constant},{T ∈ SΦ(E,F):either dim N(T)=constant< ∞ or codim R(T)=constant< ∞} and {T ∈ R(E,F):Rank T=constant< ∞}. Then it is known that Σ is a smooth submanifold of B(E,F) with the tangent space TAΣ={B ∈ B(E,F):BN(A)-R(A) } for any A ∈Σ. However,for ...     

On symmetrization of jets

Let F = F ^(A,H,t) and F¹ = F^(A1,H1,t1){F^1} = {F^{({A^1},{H^1},{t^1})}} be fiber product preserving bundle functors on the category FM _m of fibred manifolds Y with m-dimensional bases and fibred maps covering local diffeomorphisms. We define a quasi-morphism (A, H, t) → (A ¹, H ¹, t ¹) to be a GL(m)-invariant algebra homomorphism ν: A → A ¹ with t ¹ = ν ∘ t. The main result is that there exists an FM _m -natural transformation FY → F ¹ Y depending on a classical linear connection on the base of Y if and only if there exists a quasi-morphism (A, H, t) → (A ¹, H ¹, t ¹). As applications, we study existence problems of symmetrization (holonomization) of higher order jets and of holonomic prolongation of general connections.     

On linear operators strongly preserving invariants of Boolean matrices

Let U _k be the general Boolean algebra and T a linear operator on M _m,n (U _k ). If for any A in M _m,n (U _k ) (M _n (U _k ), respectively), A is regular (invertible, respectively) if and only if T(A) is regular (invertible, respectively), then T is said to strongly preserve regular (invertible, respectively) matrices. In this paper, we will give complete characterizations of the linear operators that strongly preserve regular (invertible, respectively) matrices over U _k . Meanwhile, noting that a general Boolean algebra U _k is isomorphic to a finite direct product of binary Boolean algebras, we also give some characterizations of linear operators that strongly preserve regular (invertible, respectively) matrices over 169-7 _k from another point of view.     

On Representations Associated with Completely n-Positive Linear Maps on Pro-C^＊-Algebras

It is shown that an n × n matrix of continuous linear maps from a pro-C^＊-algebra A to L（H）, which verifies the condition of complete positivity, is of the form [V^＊TijФ（·）V]^n i,where Ф is a representation of A on a Hilbert space K, V is a bounded linear operator from H to K, and j=1,[Tij]^n i,j=1 is a positive element in the C^＊-algebra of all n×n matrices over the commutant of Ф（A） in L（K）. This generalizes a result of C. Y.Suen in Proc. Amer. Math. Soc., 112（3）, 1991, 709-712. Also, a covariant version of this construction is given.     

Maps preserving square-zero matrices on the algebra consisting of all upper triangular matrices

Let F be a field, T _n (F) (respectively, N _n (F)) the matrix algebra consisting of all n × n upper triangular matrices (respectively, strictly upper triangular matrices) over F. A ∈ T _n (F) is said to be square zero if A ² = 0. In this article, we firstly characterize non-singular linear maps on N _n (F) preserving square-zero matrices in both directions, then by using it we determine non-singular linear maps on T _n (F) preserving square-zero matrices in both directions.     

Invertibility of Matrices Over Subrings

Given rings R ? S, consider the division closure 𝒟(R, S) and the rational closure ?(R, S) of R in S. If S is commutative, then 𝒟(R, S) = ?(R, S) = RT ^?1, where T = {t ∈ R | t ^?1 ∈ S}. We show that this is also true if we assume only that R is commutative.     

On a criterion for the complete controllability of nonautonomous linear systems

We describe the controllability sets of linear nonautonomous systems ẋ = A(t)x + B(t)u, x ∈ ℝ ⁿ , u ∈ U ⊆ ℝ ^m , with entire matrix functions A(t) and B(t) and with a linear set U of control constraints. We derive a criterion for the complete controllability of these linear systems in terms of derivatives of the entire matrix functions A(t) and B(t) at zero. This complete controllability criterion is compared with the Kalman and Krasovskii criteria.     

Sign change of the function <Emphasis Type="Italic">S</Emphasis>(<Emphasis Type="Italic">t</Emphasis>) on short intervals

A theorem for the sign variation of the argument of the Riemann zeta function S(t) in the interval (t − A, t + A) with A = 4.39 ln ln ln ln T for each t, T ≤ t ≤ T + H excluding values from the set E with the measure mes(E) = O(H(ln ln T)⁻¹(ln ln ln T)^−0,5) is proved.