共查询到20条相似文献,搜索用时 421 毫秒
1.
Thierry Gaspari 《Comptes Rendus Mathematique》2002,334(3):189-194
We study the range of the derivative of a Frechet differentiable bump. X is an infinite dimensional separable Cp-smooth Banach space. We first prove that any connected open subset of containing 0 is the range of the derivative of a Cp-bump. Next, analytic subsets of which satisfy a natural linkage condition are the range of the derivative of a C1-bump. We find analogues of these results in finite dimensions. We finally show that is the closure of its interior, if f is a C2-bump on . To cite this article: T. Gaspari, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 189–194. 相似文献
2.
Let A be a C1-algebra and X a Banach A-module. The module action of A on X gives rise to module actions of on and , and derivations of A into X (resp. ) extend to derivations of into (resp. ). If A is nuclear, and X is a dual Banach A-module with weakly sequentially complete, then every derivation of A into X is inner. Under the same hypothesis on A, the extension to the finite part of of any derivation of A into any dual Banach A-module is inner, as are all derivations of A into . Every derivation of a semifinite von Neumann algebra into its predual is inner. 相似文献
3.
Palle E.T Jørgensen 《Journal of Functional Analysis》1975,20(2):105-135
In this paper we apply the theory of second-order partial differential operators with nonnegative characteristic form to representations of Lie groups. We are concerned with a continuous representation U of a Lie group G in a Banach space . Let be the enveloping algebra of G, and let dU be the infinitesimal homomorphism of into operators with the Gårding vectors as a common invariant domain. We study elements in of the form with the Xj,'s in the Lie algebra .If the elements X0, X1,…, Xr generate as a Lie algebra then we show that the space of C∞-vectors for U is precisely equal to the C∞-vectors for the closure . This result is applied to obtain estimates for differential operators.The operator is the infinitesimal generator of a strongly continuous semigroup of operators in . If X0 = 0 we show that this semigroup can be analytically continued to complex time ζ with Re ζ > 0. The generalized heat kernels of these semigroups are computed. A space of rapidly decreasing functions on G is introduced in order to treat the heat kernels.For unitary representations we show essential self-adjointness of all operators with X0 in the real linear span of the Xj's. An application to quantum field theory is given.Finally, the new characterization of the C∞-vectors is applied to a construction of a counterexample to a conjecture on exponentiation of operator Lie algebras.Our results on semigroups of exponential growth, and on the space of C∞ vectors for a group representation can be viewed as generalizations of various results due to Nelson-Stinespring [18], and Poulsen [19], who prove essential self-adjointness and a priori estimates, respectively, for the sum of the squares of elements in a basis for (the Laplace operator). The work of Hörmander [11] and Bony [3] on degenerate-elliptic (hypoelliptic) operators supplies the technical basis for this generalization. The important feature is that elliptic regularity is too crude a tool for controlling commutators. With the aid of the above-mentioned hypoellipticity results we are able to “control” the (finite dimensional) Lie algebra generated by a given set of differential operators. 相似文献
4.
Let 1 < p ? 2 ? q < ∞ and X be either a Banach lattice which is p-convex and q-concave or a unitary ideal of operators on l2 which is modeled on a symmetric space which is p-convex and q-concave. If E ?X is any n-dimensional subspace, then both the distance from E to and the relative projection constant of E in X are dominated by . 相似文献
5.
A new method of finding explicit solutions of Lyapunov equations is described based on a lemma on one-dimensional perturbations of invertible operators. If Y satisfies the equation for an appropriate vector b, then X = Y-1 satisfies for a given vector a. A concrete example [with a=(1,0,…,0)T] is given. 相似文献
6.
Larry W. Cusick 《Topology and its Applications》1985,21(1):9-18
We show that if X is a finite CW-complex admitting a fixed point free involution then there is a singly graded spectral sequence with and . As an application we prove that for any n > 0 there is a natural number k(n) such that if n > k(n) and X is a homotopy , then X will not admit a fixed point free involution. 相似文献
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For a class of subsets of a set X, let V() be the smallest n such that no n-element set F?X has all its subsets of the form A ∩ F, A ∈ . The condition V() <+∞ has probabilistic implications. If any two-element subset A of X satisfies both A ∩ C = Ø and A ? D for some C, D∈, then if and only if is linearly ordered by inclusion. If is of the form , i=1,2,…,n}, where each is linearly ordered by inclusion, then . If H is an (n-1)-dimensional affine hyperplane in an n-dimensional vector space of real functions on X, and is the collection of all sets {x: f(x)>0} for f in H, then . 相似文献
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12.
Hermann König 《Journal of Functional Analysis》1977,24(1):32-51
For an open set Ω ? N, 1 ? p ? ∞ and λ ∈ +, let denote the Sobolev-Slobodetzkij space obtained by completing in the usual Sobolev-Slobodetzkij norm (cf. A. Pietsch, “r-nukleare Sobol. Einbett. Oper., Ellipt. Dgln. II,” Akademie-Verlag, Berlin, 1971, pp. 203–215). Choose a Banach ideal of operators , 1 ? p, q ? ∞ and a quasibounded domain Ω ? N. Theorem 1 of the note gives sufficient conditions on λ such that the Sobolev-imbedding map exists and belongs to the given Banach ideal : Assume the quasibounded domain fulfills condition Ckl for some l > 0 and 1 ? k ? N. Roughly this means that the distance of any to the boundary ?Ω tends to zero as for , and that the boundary consists of sufficiently smooth ?(N ? k)-dimensional manifolds. Take, furthermore, 1 ? p, q ? ∞, p > k. Then, if μ, ν are real positive numbers with λ = μ + v ∈ , μ > λ S(; p,q:N) and v > N/l · λD(;p,q), one has that belongs to the Banach ideal . Here λD(;p,q;N)∈+ and λS(;p,q;N)∈+ are the D-limit order and S-limit order of the ideal , introduced by Pietsch in the above mentioned paper. These limit orders may be computed by estimating the ideal norms of the identity mappings lpn → lqn for n → ∞. Theorem 1 in this way generalizes results of R. A. Adams and C. Clark for the ideals of compact resp. Hilbert-Schmidt operators (p = q = 2) as well as results on imbeddings over bounded domains.Similar results over general unbounded domains are indicated for weighted Sobolev spaces.As an application, in Theorem 2 an estimate is given for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in , where Ω fulfills condition C1l.For an open set Ω in N, let denote the Sobolev-Slobodetzkij space obtained by completing in the usual Sobolev-Slobodetzkij norm, see below. Taking a fixed Banach ideal of operators and 1 ? p, q ? ∞, we consider quasibounded domains Ω in N and give sufficient conditions on λ such that the Sobolev imbedding operator exists and belongs to the Banach ideal. This generalizes results of C. Clark and R. A. Adams for compact, respectively, Hilbert-Schmidt operators (p = q = 2) to general Banach ideals of operators, as well as results on imbeddings over bounded domains. Similar results over general unbounded domains may be proved for weighted Sobolev spaces. As an application, we give an estimate for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in , where Ω is a quasibounded open set in N. 相似文献
13.
The following estimate of the pth derivative of a probability density function is examined: , where hk is the kth Hermite function and Σi = 1nhk(p)(Xi) is calculated from a sequence X1,…, Xn of independent random variables having the common unknown density. If the density has r derivatives the integrated square error converges to zero in the mean and almost completely as rapidly as O(n?α) and O(n?α log n), respectively, where . Rates for the uniform convergence both in the mean square and almost complete are also given. For any finite interval they are O(n?β) and , respectively, where . 相似文献
14.
Fredric D. Ancel 《Topology and its Applications》1983,15(2):99-110
A collection of proper maps into a locally compact Hausdorff space X fixes the topology of X if the only locally compact Hausdorff topology on X which makes each element of continuous and proper is the given topology. In I2=[-1, 1]×[-1, 1], neither the collection of analytic paths nor the collection of regular twice differentiable paths fixes the topology. However, in I2, both the collection of C∞ arcs and the collection of regular C1 arcs fix the topology. In , the collection of regular injective entire rays together with either the collection of C∞ arcs or the collection of regular C1 arcs fixes the topology. 相似文献
15.
Ivan Singer 《Journal of Mathematical Analysis and Applications》1980,76(2):339-368
We show that, if (F →uX) is a linear system, a convex target set and a convex functional, then, under suitable assumptions, the computation of inf ) can be reduced to the computation of the infimum of h on certain strips or hyperplanes in F, determined by elements of , or of the infima on F of Lagrangians, involving elements of . Also, we prove similar results for a convex system (F →uX) and the convex cone Ω of all non-positive elements in X. 相似文献
16.
The main result is that every weakly compact operator between Banach spaces factors through a reflexive Banach space. Applications of the result and technique of proof include new results (e.g., separable conjugate spaces embed isomorphically in spaces with boundedly complete bases; convex weakly compact sets are affinely homeomorphic to sets in a reflexive space) and simple proofs of known results (e.g., there is a reflexive space failing the Banach-Saks property; if X is separable, then for some Z; there is a separable space which does not contain l1 whose dual is nonseparable). 相似文献
17.
Shlomo Moran 《Journal of Combinatorial Theory, Series B》1984,37(2):113-141
Let V be a set of n points in Rk. Let d(V) denote the diameter of V, and l(V) denote the length of the shortest circuit which passes through all the points of V. (Such a circuit is an “optimal TSP circuit”.) lk(n) are the extremal values of l(V) defined by lk(n)=max{l(V)|V∈Vnk}, where Vnk={V|V?Rk,|V|=n, d(V)=1}. A set V∈Vnk is “longest” if l(V)=lk(n). In this paper, first some geometrical properties of longest sets in R2 are studied which are used to obtain l2(n) for small n′s, and then asymptotic bounds on lk(n) are derived. Let δ(V) denote the minimal distance between a pair of points in V, and let: δk(n)=max{δ(V)|V∈Vnk}. It is easily observed that . Hence, exists. It is shown that for all , and hence, for all . For k=2, this implies that , which generalizes an observation of Fejes-Toth that . It is also shown that . The above upper bound is used to improve related results on longest sets in k-dimensional unit cubes obtained by Few (Mathematika2 (1955), 141–144) for almost all k′s. For k=2, Few's technique is used to show that . 相似文献
18.
Let (, G, α) be a dynamical system and let δ be a closed 1 derivation in which commutes with α and satisfies G ? ker(δ). If is a separable Type I algebra and G is a second countable compact group, then δ generates a strongly continuous one parameter group of 1 automorphisms of . 相似文献
19.
Suppose A, D1,…,Dm are n × n matrices where A is self-adjoint, and let . It is shown that if , then the spectrum of X is majorized by the spectrum of A. In general, without assuming any condition on D1,…,Dm, a result is obtained in terms of weak majorization. If each Dk is a diagonal matrix, then X is equal to the Schur (entrywise) product of A with a positive semidefinite matrix. Thus the results are applicable to spectra of Schur products of positive semidefinite matrices. If A, B are self-adjoint with B positive semidefinite and if bii = 1 for each i, it follows that the spectrum of the Schur product of A and B is majorized by that of A. A stronger version of a conjecture due to Marshall and Olkin is also proved. 相似文献
20.
Simon Wassermann 《Journal of Functional Analysis》1976,23(3):239-254
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 , 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 is the group C1-algebra of F2, there is a canonical homomorphism λl of onto . The principal result of this paper is that there is a norm ζ on , distinct from α, relative to which the homomorphism is bounded ( 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 {}.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. 相似文献