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1.
Let (T, Σ, μ) be a measure space, E a Banach space, and Lp(E, μ) the Lebesque-Bochner function spaces for 1 < p < ∞. It is shown that Lp(E, μ) is smooth if and only if E is smooth. From this result a Radon-Nikodym theorem for conjugates of smooth Banach spaces is established, and thus a general geometric condition on E sufficient to ensure that for all p, 1 < p < ∞. Alternate proofs of certain known results concerning the duals of Lp(E, μ) spaces are provided. 相似文献
2.
We give several characterizations of those Banach spaces X such that the dual contains a complemented subspace isomorphic to . We investigate operators on separable ∞ spaces whose adjoints have nonseparable ranges and apply our results to obtain a structure theorem for ∞ spaces whose duals are not isomorphic to l1(Γ). 相似文献
3.
Roger Howe 《Journal of Functional Analysis》1979,32(3):297-303
Let (i, H, E) and (j, K, F) be abstract Wiener spaces and let α be a reasonable norm on E ? F. We are interested in the following problem: is () an abstract Wiener space ? The first thing we do is to prove that the setting of the problem is meaningfull: namely, i ? j is always a continuous one to one map from into . Then we exhibit an example which shows that the answer cannot be positive in full generality. Finally we prove that if F=Lp(X,,λ) for some σ-finite measure λ ? 0 then (X,,λ) is an abstract Wiener space. By-products are some new results on γ-radonifying operators, and new examples of Banach spaces and cross norms for which the answer is affirmative (in particular α = π the projective norm, and F=L1(X,,λ)). 相似文献
4.
Wolfgang Ruess 《Journal of Mathematical Analysis and Applications》1981,84(2):400-417
This is a study of compactness in (a) spaces Kb(X, Y) of compact linear operators, (b) injective tensor products , and (c) spaces Lc(X, Y) of continuous linear operators, and its various relationships with equicontinuity and collective compactness. Among the applications is a result on factoring compact sets of compact operators compactly and uniformly through one and the same reflexive Banach space. 相似文献
5.
In [6, theorem IV.8.18], relatively norm compact sets K in Lp(μ) are characterized by means of strong convergence of conditional expectations, Eπf → f in Lp(μ), uniformly for f ∈ K, where (Eπ) is the family of conditional expectations corresponding to the net of all finite measurable partitions.In this paper we extend the above result in several ways: we consider nets of not necessarily finite partitions; we consider spaces of vector valued pth power Bochner integrable functions (and spaces M(Σ, E) of vector valued measures with finite variation); we characterize relatively strong compact sets K in by means of uniform strong convergence Eπf → f, as well as relatively weak compact sets K by means of uniform weak convergence Eπf → f. Previously, in [4], uniform strong convergence (together with some other conditions) was proved to be sufficient (but not necessary) for relative weak compactness. 相似文献
6.
Nonlinear partial differential operators having the form G(u) = g(u, D1u,…, DNu), with g?C(R × RN), are here shown to be precisely those operators which are local, (locally) uniformly continuous on, , and (roughly speaking) translation invariant. It is also shown that all such partial differential operators are necessarily bounded and continuous with respect to the norm topologies of . 相似文献
7.
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
8.
Rahman Younis 《Journal of Functional Analysis》1981,44(3):381-387
The main result of this paper is that if F is a closed subset of the unit circle, then is an M-ideal of . Consequently, if then ? has a closest element in H∞ + LF∞. Furthermore, if is not the dual of any Banach space. 相似文献
9.
Let (m?n) denote the linear space of all m × n complex or real matrices according as = or . Let c=(c1,…,cm)≠0 be such that c1???cm?0. The c-spectral norm of a matrix A?m×n is the quantity . where σ1(A)???σm(A) are the singular values of A. Let d=(d1,…,dm)≠0, where d1???dm?0. We consider the linear isometries between the normed spaces and , and prove that they are dual transformations of the linear operators which map (d) onto (c), where . 相似文献
10.
Jean Bourgain 《Comptes Rendus Mathematique》2002,335(6):529-531
We consider quasi-periodic Schrödinger operators H on of the form H=Hλ,x,ω=λv(x+nω)δn,n′+Δ where v is a non-constant real analytic function on the d-torus and Δ denotes the discrete lattice Laplacian on . Denote by Lω(E) the Lyapounov exponent, considered as function of the energy E and the rotation vector . It is shown that for |λ|>λ0(v), there is the uniform minoration for all E and ω. For all λ and ω, Lω(E) is a continuous function of E. Moreover, Lω(E) is jointly continuous in (ω,E), at any point such that k·ω0≠0 for all . To cite this article: J. Bourgain, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 529–531. 相似文献
11.
Alain A. Lewis 《Mathematical Social Sciences》1985,9(3):197-247
Let 1M be a denumerately comprehensive enlargement of a set-theoretic structure sufficient to model R. If F is an internal 1finite subset of 1N such that , we define a class of 1finite cooperative games having the form , where A(F) is the internal algebra of the internal subsets of F, and is a set-function with , , and . If is the space of S-imputations of a game ΓF(1ν) such that , for some , then we prove that contains two nonempty subsets: and , termed the quasi-kernel and S-bargaining set, respectively. Both and are external solution concepts for games of the form ΓF (1ν) and are defined in terms of predicates that are approximate in infinitesimal terms. Furthermore, if L(Θ) is the Loeb space generated by the 1finitely additive measure space 〈F, A(F), UF〉, and if a game ΓF(1ν) has a nonatomic representation on L(Θ) with respect to S-bounded transformations, then the standard part of any element in is Loeb-measurable and belongs to the quasi-kernel of defined in standard terms. 相似文献
12.
Herbert Halpern 《Journal of Functional Analysis》1980,36(3):313-342
Let be a von Neumann algebra, let σ be a strongly continuous representation of the locally compact abelian group G as 1-automorphisms of . Let M(σ) be the Banach algebra of bounded linear operators on generated by ∝ σtdμ(t) (μ?M(G)). Then it is shown that M(σ) is semisimple whenever either (i) 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 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 . Also the spectral subspaces of σ are given in terms of projections. 相似文献
13.
Woody Lichtenstein 《Journal of Functional Analysis》1979,34(3):433-455
For a symmetric space of compact type, the highest-weight vectors for representations of G occurring in become heavily concentrated near certain submanifolds of as the highest weight goes to infinity. This fact is applied to obtain estimates for the spectral measures of the operators qλ = PλqPλ, where is an orthogonal projection onto a G-irreducible summand, and q: G/K → is a continuous function acting on by multiplication. 相似文献
14.
Let K1 and K2 be number fields and . Suppose and are of prime degree p but are not necessarily normal. Let N1 and N2 be the normal closures of K1 and K2 over F, respectively, L = K1K2, N = N1N2, and be a prime divisor of N which divides p and is totally ramified in and . Let be the ramification index of in , be the total ramification number of in , and . Then (K1, K2) is exactly divisible by M, where . 相似文献
15.
Alladi Sitaram 《Journal of Functional Analysis》1978,27(2):179-184
Let G be a semisimple noncompact Lie group with finite center and let K be a maximal compact subgroup. Then W. H. Barker has shown that if T is a positive definite distribution on G, then T extends to Harish-Chandra's Schwartz space 1(G). We show that the corresponding property is no longer true for the space of double cosets . If G is of real-rank 1, we construct liner functionals for each p, 0 < p ? 2, such that but Tp does not extend to a continuous functional on . In particular, if p ? 1, Tv does not extend to a continuous functional on . We use this to answer a question (in the negative) raised by Barker whether for a K-bi-invariant distribution T on G to be positive definite it is enough to verify that . The main tool used is a theorem of Trombi-Varadarajan. 相似文献
16.
Yuh-Jia Lee 《Journal of Functional Analysis》1982,47(2):153-164
Let (H, B) be an abstract Wiener pair and pt the Wiener measure with variance t. Let a be the class of exponential type analytic functions defined on the complexification [B] of B. For each pair of nonzero complex numbers α, β and f ? a, we define We show that the inverse α,β?1 exists and there exist two nonzero complex numbers α′,β′ such that . Clearly, the Fourier-Wiener transform, the Fourier-Feynman transform, and the Gauss transform are special cases of α,β. Finally, we apply the transform to investigate the existence of solutions for the differential equations associated with the operator c, where c is a nonzero complex number and c is defined by where Δ is the Laplacian and (·, ·) is the pairing. We show that the solutions can be represented as integrals with respect to the Wiener measure. 相似文献
17.
Derek W Robinson 《Journal of Functional Analysis》1977,24(3):280-290
Let U, V be two strongly continuous one-parameter groups of bounded operators on a Banach space with corresponding infinitesimal generators S, T. We prove the following: ∥Ut, ? Vt ∥ = O(t), t → 0, if and only if U = V; ∥Ut ? Vt∥ = O(tα), t → 0; with 0 ? α ? 1, if and only if , where Ω, P, are bounded operators on such that if and only if has a bounded extension to 1. Further results of this nature are inferred for semigroups, reflexive spaces, Hilbert spaces, and von Neumann algebras. 相似文献
18.
Let be a hyperconvex domain. Denote by the class of negative plurisubharmonic functions ? on with boundary values 0 and finite Monge–Ampère mass on Then denote by the class of negative plurisubharmonic functions ? on for which there exists a decreasing sequence (?)j of plurisubharmonic functions in converging to ? such that It is known that the complex Monge–Ampère operator is well defined on the class and that for a function the associated positive Borel measure is of bounded mass on A function from the class is called a plurisubharmonic function with bounded Monge–Ampère mass on We prove that if and are hyperconvex domains with and there exists a plurisubharmonic function such that on and Such a function is called a subextension of ? to From this result we deduce a global uniform integrability theorem for the classes of plurisubharmonic functions with uniformly bounded Monge–Ampère masses on To cite this article: U. Cegrell, A. Zeriahi, C. R. Acad. Sci. Paris, Ser. I 336 (2003). 相似文献
19.
Ming-Po Chen Cheh-Chih Yeh Cheng-Shu Yu 《Journal of Mathematical Analysis and Applications》1977,59(2):211-215
For nonlinear retarded differential equations and the sufficient conditions are given on fi, pi, Fi, and h under which every bounded nonoscillatory solution of () or () tends to zero as t → ∞. 相似文献
20.
Milton Rosenberg 《Journal of multivariate analysis》1978,8(2):295-316
Let p, q be arbitrary parameter sets, and let be a Hilbert space. We say that x = (xi)i?q, xi ? , is a bounded operator-forming vector (?Fq) if the Gram matrix 〈x, x〉 = [(xi, xj)]i?q,j?q is the matrix of a bounded (necessarily ≥ 0) operator on , the Hilbert space of square-summable complex-valued functions on q. Let A be p × q, i.e., let A be a linear operator from to . Then exists a linear operator ǎ from (the Banach space) Fq to Fp on (A) = {x:x ? Fq, is p × q bounded on } such that y = ǎx satisfies yj?σ(x) = {space spanned by the xi}, 〈y, x〉 = A〈x, x〉 and . This is a generalization of our earlier [J. Multivariate Anal.4 (1974), 166–209; 6 (1976), 538–571] results for the case of a spectral measure concentrated on one point. We apply these tools to investigate q-variate wide-sense Markov processes. 相似文献