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1.
Given rational matrix functions ψ1(λ) = Im + C1(λIn1 − A1)−1B1 and ψ2(λ) = Im + C2(λIn2 − 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) = Im + C(λIn − A)−1B such that WH2m⊥ = ψ 1H2m⊥and WH2m = ψ2H2m. 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. 相似文献
2.
For aC *-algebraA with a conditional expectation Φ:A → A onto a subalgebraB we have the linear decompositionA=B⊕H whereH=ker(Φ). Since Φ preserves adjoints, it is also clear that a similar decomposition holds for the selfadjoint parts:A s =B s ⊕H s (we useV s ={aεV;a *=a} for any subspaceV of A). Apply now the exponential function to each of the three termsA s ,B s , andH s . The results are: the setG + of positive invertible elements ofA, the setB + of positive invertible elements ofB, and the setC={eh;h *=h, Φ(h)=0}, respectively. We consider here the question of lifting the decompositionA s =B s ⊕H s to the exponential sets. Concretely, is every element ofG + the product of elements ofB + andC, respectively, just as any selfadjoint element ofA is the sum of selfadjoint elements ofB andH? The answer is yes in the following sense: Eacha ε G + is the positive part of a productbe of elementsb ε B + and c εC, and bothb andc are uniquely determined and depend analytically ona. This can be rephrased as follows: The map (6, c) →(bc) + is an analytic diffeomorphism fromB + x C ontoG +, where for any invertiblex ε A we denote with x+ the positive square root ofxx *. This result can be expressed equivalently as: The map (b, c) →bcb is a diffeomorphism between the same spaces. Notice that combining the polar decomposition with these results we can write every invertibleg ε A asg=bcu, whereb ε B +,c ε C, andu is unitary. This decomposition is unique and the factorsb, c, u depend analytically ofg. In the case of matrix algebras with Φ=trace/dimension, the factorization corresponds tog=| det(g)|cu withc > 0,det(c)=1, andu unitary. This paper extends some results proved by G. Corach and the authors in [2]. Also, Theorem 2 states that the reductive homogeneous space resulting from a conditional expectation satisfies the regularity hypothesis introduced by L. Mata-Lorenzo and L. Recht in [5], Definition 11.1. The situation considered here is the ”general context” for regularity indicated in the introduction of the last mentioned paper. 相似文献
3.
Given operator polynomials A(λ) and B(λ), one of which is monic, conditions are given for the existence of operator polynomials C(λ) and D(λ) such that A(λ)C(λ) + B(λ)D(λ) = I for every . A special case will give a characterization of controlla- bility of an infinite-dimensional linear control system. 相似文献
4.
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. 相似文献
5.
Abbas Salemi 《Mathematische Nachrichten》2000,217(1):141-146
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. 相似文献
6.
Ryuzaburo Noda 《Journal of Combinatorial Theory, Series A》1978,24(1):89-95
t?(2k, k, λ) designs having a property similar to that of Hadamard 3-designs are studied. We consider conditions (i), (ii), or (iii) for t?(2k, k, λ) designs: (i) The complement of each block is a block. (ii) If A and B are a complementary pair of blocks, then ∥ A ∩ C ∥ = ∥ B ∩ C ∥ ± u holds for any block C distinct from A and B, where u is a positive integer. (iii) if A and B are a complementary pair of blocks, then ∥ A ∩ C ∥ = ∥ B ∩ C ∥ or ∥ A ∩ C ∥ = ∥ B ∩ C ∥ ± u holds for any block C distinct from A and B, where u is a positive integer. We show that a t?(2k, k, λ) design with t ? 2 and with properties (i) and (ii) is a 3?(2u(2u + 1), u(2u + 1), u(2u2 + u ? 2)) design, and that a t?(2k, k, λ) design with t ? 4 and with properties (i) and (iii) is the 5-(12, 6, 1) design, the 4-(8, 4, 1) design, a design, or a design. 相似文献
7.
We consider transfer operators acting on spaces of holomorphic functions, and provide explicit bounds for their eigenvalues. More precisely, if Ω is any open set in Cd, and L is a suitable transfer operator acting on Bergman space A2(Ω), its eigenvalue sequence {λn(L)} is bounded by |λn(L)|?Aexp(−an1/d), where a,A>0 are explicitly given. 相似文献
8.
LetB denote the infinitesimal operator of a strongly continuous semigroup S(t), with resolvent Rλ, on Banach space L. We define related operators P and V so that λRλf = Pf + λVf + o(λ), as λ → 0+. For α, η > 0 and possibly unbounded, linear operator A, we let Uα, η(t) represent a strongly continuous semigroup generated by αA + ηB. We show that under appropriate simultaneous convergence of α and η, Uα, η(t) converges strongly to a strongly continous semigroup U(t), having infinitesimal operator characterized through PA(VA)rf where r =min{j ? 0, PA(VA)j ≠ 0}. We apply the abstract perturbation theorem to a singular perturbation initial-value problem, of Tihonov-type, for a non-linear system of ordinary differential equations. 相似文献
9.
Jiankui Li 《Linear algebra and its applications》2010,432(1):5-322
For a commutative subspace lattice L in a von Neumann algebra N and a bounded linear map f:N∩algL→B(H), we show that if Af(B)C=0 for all A,B,C∈N∩algL satisfying AB=BC=0, then f is a generalized derivation. For a unital C∗-algebra A, a unital Banach A-bimodule M, and a bounded linear map f:A→M, we prove that if f(A)B=0 for all A,B∈A with AB=0, then f is a left multiplier; as a consequence, every bounded local derivation from a C∗-algebra to a Banach A-bimodule is a derivation. We also show that every local derivation on a semisimple free semigroupoid algebra is a derivation and every local multiplier on a free semigroupoid algebra is a multiplier. 相似文献
10.
In this paper the rate of stability of solutions of matrix polynomial equations of the typeA 0+A 1 X+A 2 X 2+...+A m X m =0 is studied. Particular attention is given to the case where the matrix polynomialL(λ):=A 0+A 1 λ+A 2 λ 2+...+A m λ m is weakly hyperbolic, i.e., for every non-zero vectorx the scalar polynomial 〈L(λ)x, x〉 has only real roots. Also the rate of stability of solutions of matrix quadratic equations of the typeXBX+XA-DX-C=0 is studied. Here the special case that is of interest to continuous-time optimal control theory, that is, the case whereB=B * is positive semidefinite andC=C *,A=?D *, is discussed in detail. The analogous theory for the discrete-time optimal control leads to the equation $$X = A^* XA + Q - (B^* XA)^* (R + B^* XB)^{ - 1} B^* XA,$$ and the rate of stability of solutions of this equation is also studied. Most of the problems are discussed in both real and complex settings. 相似文献
11.
R.C. Thompson 《Linear algebra and its applications》1976,14(2):135-177
Given n-square Hermitian matrices A,B, let Ai,Bi 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(μAi?λBi) 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. 相似文献
12.
L.E. Ward 《Topology and its Applications》1985,20(1):47-58
Let X be a Peano continuum, C(X) its space of subcontinua, and C(X, ε) the space of subcontinua of diameter less than ε. A selection on some subspace of C(X) is a continuous choice function; the selection σ is rigid if σ(A) ? B ? A implies σ(A) = σ(B). It is shown that X is a local dendrite (contains at most one simple closed curve) if and only if there exists ε > 0 such that C(X, ε) admits a selection (rigid selection). Further, C(X) admits a local selection at the subcontinuum A if and only if A has a neighborhood (relative to the space C(X)) which contains no cyclic local dendrite; moreover, that local selection may be chosen to be a constant. 相似文献
13.
B.P. Duggal 《Journal of Mathematical Analysis and Applications》2010,370(2):584-587
Let B(H) denote the algebra of operators on an infinite dimensional complex Hilbert space H, and let A○∈B(K) denote the Berberian extension of an operator A∈B(H). It is proved that the set theoretic function σ, the spectrum, is continuous on the set C(i)⊂B(Hi) of operators A for which σ(A)={0} implies A is nilpotent (possibly, the 0 operator) and at every non-zero λ∈σp(A○) for some operators X and B such that λ∉σp(B) and σ(A○)={λ}∪σ(B). If CS(m) denotes the set of upper triangular operator matrices , where Aii∈C(i) and Aii has SVEP for all 1?i?m, then σ is continuous on CS(m). It is observed that a considerably large number of the more commonly considered classes of Hilbert space operators constitute sets C(i) and have SVEP. 相似文献
14.
Frank Uhlig 《Linear algebra and its applications》1973,7(4):281-289
The simultaneous diagonalization of two real symmetric (r.s.) matrices has long been of interest. This subject is generalized here to the following problem (this question was raised by Dr. Olga Taussky-Todd, my thesis advisor at the California Institute of Technology): What is the first simultaneous block diagonal structure of a nonsingular pair of r.s. matrices ? For example, given a nonsingular pair of r.s. matrices S and T, which simultaneous block diagonalizations X′SX = diag(A1, , Ak), X′TX = diag(B1,, Bk) with dim Ai = dim Bi and X nonsingular are possible for 1 ? k ? n; and how well defined is a simultaneous block diagonalization for which k, the number of blocks, is maximal? Here a pair of r.s. matrices S and T is called nonsingular if S is nonsingular.If the number of blocks k is maximal, then one can speak of the finest simultaneous block diagonalization of S and T, since then the sizes of the blocks Ai are uniquely determined (up to permutations) by any set of generators of the pencil P(S, T) = {aS + bT|a, tb ε R} via the real Jordan normal form of S?1T. The proof uses the canonical pair form theorem for nonsingular pairs of r.s. matrices. The maximal number k and the block sizes dim Ai are also determined by the factorization over C of ? (λ, μ) = det(λS + μT) for λ, μ ε R. 相似文献
15.
Young Han Choe 《Journal of Mathematical Analysis and Applications》1985,106(2):293-320
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 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 limk → 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 is obtained. 相似文献
16.
Mihály Weiner 《Linear algebra and its applications》2010,433(3):520-533
In this work it is shown that certain interesting types of orthogonal system of subalgebras (whose existence cannot be ruled out by the trivial necessary conditions) cannot exist. In particular, it is proved that there is no orthogonal decomposition of Mn(C)⊗Mn(C)≡Mn2(C) into a number of maximal abelian subalgebras and factors isomorphic to Mn(C) in which the number of factors would be 1 or 3.In addition, some new tools are introduced, too: for example, a quantity c(A,B), which measures “how close” the subalgebras A,B⊂Mn(C) are to being orthogonal. It is shown that in the main cases of interest, c(A′,B′) - where A′ and B′ are the commutants of A and B, respectively - can be determined by c(A,B) and the dimensions of A and B. The corresponding formula is used to find some further obstructions regarding orthogonal systems. 相似文献
17.
Kenneth J Palmer 《Journal of Differential Equations》1979,33(1):16-25
Let A be a selft-adjoint operator on the Hilbert space L2Ω, ?) = {u ε Lloc2(Ω)|∫Ω|2 ?(x)dx < + ∞} defined by means of a closed, semibounded, sesquilinear form a(·, ·). We obtain a necessary and sufficuents condition for the spectrum of A to be discrete. We apply this result to a Sturm-Liouville problem for an elliptic operator with discontinuous coefficients defined on an unbounded domain and to the study of the spectrum of a Hamiltonian defined by a pseudodifferential operator. 相似文献
18.
Attila B. von Keviczky Richard L. Hall 《Journal of Mathematical Analysis and Applications》2004,292(1):274-293
The Friedrichs extension for the generalized spiked harmonic oscillator given by the singular differential operator −d2/dx2+Bx2+Ax−2+λx−α (B>0, A?0) in L2(0,∞) is studied. We look at two different domains of definition for each of these differential operators in L2(0,∞), namely C0∞(0,∞) and D(T2,F)∩D(Mλ,α), where the latter is a subspace of the Sobolev space W2,2(0,∞). Adjoints of these differential operators on C0∞(0,∞) exist as result of the null-space properties of functionals. For the other domain, convolutions and Jensen and Minkowski integral inequalities, density of C0∞(0,∞) in D(T2,F)∩D(Mλ,α) in L2(0,∞) lead to the other adjoints. Further density properties C0∞(0,∞) in D(T2,F)∩D(Mλ,α) yield the Friedrichs extension of these differential operators with domains of definition D(T2,F)∩D(Mλ,α). 相似文献
19.
Wen Zhang 《Linear algebra and its applications》2011,435(6):1326-1335
Let A and B be (not necessarily unital or closed) standard operator algebras on complex Banach spaces X and Y, respectively. For a bounded linear operator A on X, the peripheral spectrum σπ(A) of A is the set σπ(A)={z∈σ(A):|z|=maxω∈σ(A)|ω|}, where σ(A) denotes the spectrum of A. Assume that Φ:A→B is a map the range of which contains all operators of rank at most two. It is shown that the map Φ satisfies the condition that σπ(BAB)=σπ(Φ(B)Φ(A)Φ(B)) for all A,B∈A if and only if there exists a scalar λ∈C with λ3=1 and either there exists an invertible operator T∈B(X,Y) such that Φ(A)=λTAT-1 for every A∈A; or there exists an invertible operator T∈B(X∗,Y) such that Φ(A)=λTA∗T-1 for every A∈A. If X=H and Y=K are complex Hilbert spaces, the maps preserving the peripheral spectrum of the Jordan skew semi-triple product BA∗B are also characterized. Such maps are of the form A?UAU∗ or A?UAtU∗, where U∈B(H,K) is a unitary operator, At denotes the transpose of A in an arbitrary but fixed orthonormal basis of H. 相似文献
20.
Djordjije Vujadinović 《Integral Equations and Operator Theory》2013,76(2):213-224
In this paper we obtain an estimate of the norm of the Bergman projection from L p (D, dλ) onto the Besov space B p , 1 < p < + ∞. The result is asymptotically sharp when p → + ∞. Further for the case P : L 1(D, dλ) → B 1, we consider some weak type inequalities with the corresponding spaces. 相似文献