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1.
Let A be a uniform algebra on the compact space X and σ a probability measure on X. We define the Hardy spaces HP(σ) and the HP(σ) interpolating sequences S in the p-spectrum Mp of σ. Under some structural hypotheses on (A, σ), we prove that if a sequence SMp is HP(σ) interpolating, then it is Hs(σ) interpolating for s < p. In the special case of the unit ball B of ?n this answers a natural question asked in [8].  相似文献   

2.
This is the first in a series of papers on configurations in an abelian category A. Given a finite partially ordered set (I,?), an (I,?)-configuration(σ,ι,π) is a finite collection of objects σ(J) and morphisms ι(J,K) or π(J,K):σ(J)→σ(K) in A satisfying some axioms, where J,K are subsets of I. Configurations describe how an object X in A decomposes into subobjects, and are useful for studying stability conditions on A.We define and motivate the idea of configurations, and explain some natural operations upon them—subconfigurations, quotient configurations, substitution, refinements and improvements. Then we study moduli spaces of (I,?)-configurations in A, and natural morphisms between them, using the theory of Artin stacks. We prove well-behaved moduli stacks exist when A is the abelian category of coherent sheaves on a projective scheme P, or of representations of a quiver Q.In the sequels, given a stability condition (τ,T,?) on A, we will show the moduli spaces of τ-(semi)stable objects or configurations are constructible subsets in the moduli stacks of all objects or configurations. We associate infinite-dimensional algebras of constructible functions to a quiver Q using the method of Ringel-Hall algebras, and define systems of invariants of P that ‘count’ τ-(semi)stable coherent sheaves on P and satisfy interesting identities.  相似文献   

3.
We consider the question of the solvability of an inclusion F(x, σ) ∈ A, i.e., of determining a mapping (implicit function) σ ? x(σ) defined on a set such that F(x(σ), σ) ∈ A for any σ from this set. Results of this kind play a key role in the different branches of analysis and, especially, in the theory of extremal problems, where they are the main tool for deriving conditions for an extremum.  相似文献   

4.
Problems on the expansion of a semigroup and a criterion for being a Riesz basis are discussed in the present paper. Suppose that A is the generator of a C0 semigroup on a Hilbert space and σ(A)=σ1(A)∪σ2(A) with σ2(A) is consisted of isolated eigenvalues distributed in a vertical strip. It is proved that if σ2(A) is separated and for each λσ2(A), the dimension of its root subspace is uniformly bounded, then the generalized eigenvectors associated with σ2(A) form an L-basis. Under different conditions on the Riesz projection, the expansion of a semigroup is studied. In particular, a simple criterion for the generalized eigenvectors forming a Riesz basis is given. As an application, a heat exchanger problem with boundary feedback is investigated. It is proved that the heat exchanger system is a Riesz system in a suitable state Hilbert space.  相似文献   

5.
If AT(m, N), the real-valued N-linear functions on Em, and σSN, the symmetric group on {…,N}, then we define the permutation operator Pσ: T(m, N) → T(m, N) such that Pσ(A)(x1,x2,…,xN = A(xσ(1),xσ(2),…, xσ(N)). Suppose Σqi=1ni = N, where the ni are positive integers. In this paper we present a condition on σ that is sufficient to guarantee that 〈Pσ(A1?A2???Aq),A1?A2?? ? Aq〉 ? 0 for AiS(m, ni), where S(m, ni) denotes the subspace of T(m, ni) consisting of all the fully symmetric members of T(m, ni). Also we present a broad generalization of the Neuberger identity which is sometimes useful in answering questions of the type described below. Suppose G and H are subgroups of SN. We let TG(m, N) denote all AT(m, N) such that Pσ(A) = A for all σ∈G. We define the symmetrizer SG: T(m, N)→TG(m,N) such that SG(A) = 1/|G|Σσ∈G Pσ(A). Suppose H is a subgroup of G and ATH(m, N). Clearly 6SG6(A) 6? 6A6. We are interested in the reverse type of comparison. In particular, if D is a suitably chosen subset of TH(m,N), then can we explicitly present a constant C>0 such that 6 SG(A)6?C6A6 for all AD?  相似文献   

6.
Through a succession of results, it is known that if the graph of an Hermitian matrix A is a tree and if for some index j, λσ(A)∩σ(A(j)), then there is an index i such that the multiplicity of λ in σ(A(i)) is one more than that in A. We exhibit a converse to this result by showing that it is generally true only for trees. In particular, it is shown that the minimum rank of a positive semidefinite matrix with a given graph G is ?n-2 when G is not a tree. This raises the question of how the minimum rank of a positive semidefinite matrix depends upon the graph in general.  相似文献   

7.
Let A be a von Neumann algebra, let σ be a strongly continuous representation of the locally compact abelian group G as 1-automorphisms of A. Let M(σ) be the Banach algebra of bounded linear operators on A generated by ∝ σt(t) (μ?M(G)). Then it is shown that M(σ) is semisimple whenever either (i) A has a σ-invariant faithful, normal, semifinite, weight (ii) σ is an inner representation or (iii) G is discrete and each σt is inner. It is shown that the Banach algebra L(σ) generated by ∝ ?(t)σt dt (? ? L1(G)) is semisimple if a is an integrable representation. Furthermore, if σ is an inner representation with compact spectrum, it is shown that L(σ) is embedded in a commutative, semisimple, regular Banach algebra with isometric involution that is generated by projections. This algebra is contained in the ultraweakly continuous linear operators on A. Also the spectral subspaces of σ are given in terms of projections.  相似文献   

8.
We prove that a mapA εsp(σ,R), the set of infinitesimally symplectic maps, is strongly stable if and only if its centralizerC(A) insp(σ,R) contains only semisimple elements. Using the theorem that everyB insp(σ,R) close toA is conjugate by a real symplectic map to an element ofC(A), we give a new proof of the openness of the set of strongly stable maps. Then we prove that the set of strongly stable maps is the interior of the set of all infinitesimally symplectic maps with purely imaginary or zero eigenvalues, and the connected components of this set are described. Finally, we give a new proof of the analytic conjugacy theorem for an analytic curve through a given strongly stable map.  相似文献   

9.
Let B(H) denote the algebra of operators on an infinite dimensional complex Hilbert space H, and let AB(K) denote the Berberian extension of an operator AB(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 λσp(A) for some operators X and B such that λσp(B) and σ(A)={λ}∪σ(B). If CS(m) denotes the set of upper triangular operator matrices , where AiiC(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   

10.
For the lower sigma-exponent of the linear differential system ? = A(t)x, xR n , t ≥ 0, defined by the formula Δσ(A) ≡ infλ[Q]≤-σ λ 1(A + Q), σ > 0, on the basis of the lower characteristic exponents λ 1(A+Q) of perturbed linear systems with Lyapunov exponents λ[Q] ≤ ?σ < 0 of perturbations Q, we prove the following general form as a function of the parameter σ > 0. For any nondecreasing bounded function f(σ) of the parameter σ ∈ (0,+∞) that coincides with a constant on some infinite interval (σ 0,+), σ 0 ≥ 0, and satisfies the Lipschitz condition on the complementary interval (0, σ 0], we prove the existence of a linear system with coefficient matrix A f (t) bounded on the half-line [0,+∞) whose lower sigma-exponent Δσ(A f ) coincides with the function f(σ) on the entire interval (0,+∞).  相似文献   

11.
LetA be a commutativeAW*-algebra.We denote by S(A) the *-algebra of measurable operators that are affiliated with A. For an ideal I in A, let s(I) denote the support of I. Let Y be a solid linear subspace in S(A). We find necessary and sufficient conditions for existence of nonzero band preserving derivations from I to Y. We prove that no nonzero band preserving derivation from I to Y exists if either Y ? Aor Y is a quasi-normed solid space. We also show that a nonzero band preserving derivation from I to S(A) exists if and only if the boolean algebra of projections in the AW*-algebra s(I)A is not σ-distributive.  相似文献   

12.
13.
Let A and B be uniform algebras on first-countable, compact Hausdorff spaces X and Y, respectively. For fA, the peripheral spectrum of f, denoted by σπ(f)={λσ(f):|λ|=‖f‖}, is the set of spectral values of maximum modulus. A map T:AB is weakly peripherally multiplicative if σπ(T(f)T(g))∩σπ(fg)≠∅ for all f,gA. We show that if T is a surjective, weakly peripherally multiplicative map, then T is a weighted composition operator, extending earlier results. Furthermore, if T1,T2:AB are surjective mappings that satisfy σπ(T1(f)T2(g))∩σπ(fg)≠∅ for all f,gA, then T1(f)T2(1)=T1(1)T2(f) for all fA, and the map f?T1(f)T2(1) is an isometric algebra isomorphism.  相似文献   

14.
Let G be a group (or vector space) and A a group of transformations of G. A then acts as a group of transformations of P(G), the set of subsets of G. It is meaningful to study the orbit structure of P(G) under the action of A. The question of the existence of elements of P(G) with trivial isotropy subgroup seems to be of interest in studying the action of A on G. In this paper actions of affine groups over GF (2) are considered. It is proved, by an inductive construction, that every vector space over GF (2) of dimension at least six contains a subset with trivial isotropy subgroup.  相似文献   

15.
Let T be a surjective map from a unital semi-simple commutative Banach algebra A onto a unital commutative Banach algebra B. Suppose that T preserves the unit element and the spectrum σ(fg) of the product of any two elements f and g in A coincides with the spectrum σ(TfTg). Then B is semi-simple and T is an isomorphism. The condition that T is surjective is essential: An example of a non-linear and non-multiplicative unital map from a commutative C*-algebra into itself such that σ(TfTg)=σ(fg) holds for every f,g are given. We also show an example of a surjective unital map from a commutative C*-algebra onto itself which is neither linear nor multiplicative such that σ(TfTg)⊂σ(fg) holds for every f,g.  相似文献   

16.
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)={zσ(A):|z|=maxωσ(A)|ω|}, where σ(A) denotes the spectrum of A. Assume that Φ:AB 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,BA if and only if there exists a scalar λC with λ3=1 and either there exists an invertible operator TB(X,Y) such that Φ(A)=λTAT-1 for every AA; or there exists an invertible operator TB(X,Y) such that Φ(A)=λTAT-1 for every AA. If X=H and Y=K are complex Hilbert spaces, the maps preserving the peripheral spectrum of the Jordan skew semi-triple product BAB are also characterized. Such maps are of the form A?UAU or A?UAtU, where UB(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   

17.
We give sufficient conditions for a graph to have degree bounded trees. Let G be a connected graph and A a vertex subset of G. We denote by σk(A) the minimum value of the degree sum in G of any k independent vertices in A and by w(GA) the number of components in the induced subgraph GA. Our main results are the following: (i) If σk(A)≥|V(G)|−1, then G contains a tree T with maximum degree at most k and AV(T). (ii) If σkw(GA)(A)≥|A|−1, then G contains a spanning tree T such that dT(x)≤k for every xA. These are generalizations of the result by Win [S. Win, Existenz von Gerüsten mit Vorgeschriebenem Maximalgrad in Graphen, Abh. Math. Sem. Univ. Hamburg 43 (1975) 263-267] and the degree conditions are sharp.  相似文献   

18.
Let χ be a character on the symmetric group Sn, and let A = (aij) be an n-by-n matrix. The function dχ(A) = Σσ?Snχ(σ)Πnt = 1a(t) is called a generalized matrix function. If χ is an irreducible character, then dχ is called an immanent. For example, if χ is the alternating character, then dχ is the determinant, and if χ ≡ 1, then dχ is called the permanent (denoted per). Suppose that A is positive semidefinite Hermitian. We prove that the inequality (1/χ(id))dχ(A) ? per A holds for a variety of characters χ including the irreducible ones corresponding to the partitions (n ? 1,1) and (n ? 2,1,1) of n. The main technique used to prove these inequalities is to express the immanents as sums of products of principal subpermanents. These expressions for the immanents come from analogous expressions for Schur polynomials by means of a correspondence of D.E. Littlewood.  相似文献   

19.
We discuss algebraic properties of a pencil generated by two compatible Poisson tensors A(x) and B(x). From the algebraic viewpoint this amounts to studying the properties of a pair of skew-symmetric bilinear forms A and B defined on a finite-dimensional vector space. We describe the Lie group G P of linear automorphisms of the pencil P = {A + λB}. In particular, we obtain an explicit formula for the dimension of G P and discuss some other algebraic properties such as solvability and Levi-Malcev decomposition.  相似文献   

20.
Let R be a Dedekind domain, G a finite group of automorphisms of R, and A an ambiguous ideal of R i.e., σA = A for all σG. The Tate groups Hn(G, A) are considered as RG-modules. A localization theorem is proved and the precise RG-module structure determined in a particular case. In addition some remarks are made concerning cohomological triviality.  相似文献   

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