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1.
Given 1≦p<∞ and a real Banach spaceX, we define thep-absolutely summing constantμ p(X) as inf{Σ i =1/m |x*(x i)|p p Σ i =1/mx ip p]1 p}, where the supremum ranges over {x*∈X*; ‖x*‖≤1} and the infimum is taken over all sets {x 1,x 2, …,x m} ⊂X such that Σ i =1/mx i‖>0. It follows immediately from [2] thatμ p(X)>0 if and only ifX is finite dimensional. In this paper we find the exact values ofμ p(X) for various spaces, and obtain some asymptotic estimates ofμ p(X) for general finite dimensional Banach spaces. This is a part of the author’s Ph.D. Thesis prepared at the Hebrew University of Jerusalem, under the supervision of Prof. A. Dvoretzky and Prof. J. Lindenstrauss.  相似文献   

2.
We show that, whenA generates aC-semigroup, then there existsY such that [M(C)] →YX, andA| Y , the restriction ofA toY, generates a strongly continuous semigroup, where ↪ means “is continuously embedded in” and ‖x[Im(C)]≡‖C −1 x‖. There also existsW such that [C(W)] →XW, and an operatorB such thatA=B| X andB generates a strongly continuous semigroup onW. If theC-semigroup is exponentially bounded, thenY andW may be chosen to be Banach spaces; in general,Y andW are Frechet spaces. If ρ(A) is nonempty, the converse is also true. We construct fractional powers of generators of boundedC-semigroups. We would like to thank R. Bürger for sending preprints, and the referee for pointing out reference [37]. This research was supported by an Ohio University Research Grant.  相似文献   

3.
Let X, Y be complex Banach spaces. Let G be a bounded balanced domain in X and B Y be the unit ball in Y. Assume that B Y is homogeneous. Let f: GB Y be a holomorphic mapping. In this paper, we show that, if P = f(0), then we have Σ k=0D φP (P)[D k f(0)(z k )]‖/(k!‖D φP (P)‖) < 1 for z ∈ (1/3)G, where φP ∈ AutB Y ) such that φP (P) = 0. Moreover, we show that the constant 1/3 is best possible, if B Y is the unit ball of a J*-algebra. The above result was proved by Liu and Wang in the case that G = B Y is one of the four classical domains in the sense of Hua. This result generalises a classical result of Bohr.  相似文献   

4.
Let X be a normed space that satisfies the Johnson–Lindenstrauss lemma (J–L lemma, in short) in the sense that for any integer n and any x 1,…,x n X, there exists a linear mapping L:XF, where FX is a linear subspace of dimension O(log n), such that ‖x i x j ‖≤‖L(x i )−L(x j )‖≤O(1)⋅‖x i x j ‖ for all i,j∈{1,…,n}. We show that this implies that X is almost Euclidean in the following sense: Every n-dimensional subspace of X embeds into Hilbert space with distortion 22O(log*n)2^{2^{O(\log^{*}n)}} . On the other hand, we show that there exists a normed space Y which satisfies the J–L lemma, but for every n, there exists an n-dimensional subspace E n Y whose Euclidean distortion is at least 2Ω(α(n)), where α is the inverse Ackermann function.  相似文献   

5.
The bicompletion of an asymmetric normed linear space   总被引:5,自引:0,他引:5  
A biBanach space is an asymmetric normed linear space (X,‖·‖) such that the normed linear space (X,‖·‖s) is a Banach space, where ‖xs= max {‖x‖,‖-x‖} for all xX. We prove that each asymmetric normed linear space (X,‖·‖) is isometrically isomorphic to a dense subspace of a biBanach space (Y,‖·‖Y). Furthermore the space (Y,‖·‖Y) is unique (up to isometric isomorphism). This revised version was published online in August 2006 with corrections to the Cover Date.  相似文献   

6.
We investigate the completeness and completions of the normed algebras (D (1)(X), ‖ · ‖) for perfect, compact plane sets X. In particular, we construct a radially self-absorbing, compact plane set X such that the normed algebra (D (1)(X), ‖ · ‖) is not complete. This solves a question of Bland and Feinstein. We also prove that there are several classes of connected, compact plane sets X for which the completeness of (D (1)(X), ‖ · ‖) is equivalent to the pointwise regularity of X. For example, this is true for all rectifiably connected, polynomially convex, compact plane sets with empty interior, for all star-shaped, compact plane sets, and for all Jordan arcs in ℂ.  相似文献   

7.
We introduce a geometrical property of norm one complemented subspaces ofC(K) spaces which is useful for computing lower bounds on the norms of projections onto subspaces ofC(K) spaces. Loosely speaking, in the dual of such a space ifx* is a w* limit of a net (x a * ) andx*=x*1+x*2 with ‖x*‖=‖x*1‖ + ‖x*2‖, then we measure how efficiently thex a * 's can be split into two nets converging tox*1 andx*2, respectively. As applications of this idea we prove that if for everyε>0,X is a norm (1+ε) complemented subspace of aC(K) space, then it is norm one complemented in someC(K) space, and we give a simpler proof that a slight modification of anl 1-predual constructed by Benyamini and Lindenstrauss is not complemented in anyC(K) space. Research partially supported by a grant of the U.S.-Israel Binational Science Foundation. Research of the first-named author is supported in part by NSF grant DMS-8602395. Research of the second-named author was partially supported by the Fund for the Promotion of Research at the Technion, and by the Technion VPR-New York Metropolitan Research Fund.  相似文献   

8.
We give a direct, self-contained, and iterative proof that for any convex, Lipschitz andw *-lower semicontinuous function ϕ defined on aw *-compact convex setC in a dual Banach spaceX * and for any ε>0 there is anxX, with ‖x‖≤ε, such that ϕ+x attains its supremum at an extreme point ofC. This result is implicitly contained in the work of Lindenstrauss [9] and the work of Ghoussoub and Maurey on strongw *H σ sets [8]. In addition, we discuss the applications of this result to the geometry of convex sets. Research supported in part by the NSERC of Canada under grant OGP41983 for the first author and grant OGP7926 for the second author.  相似文献   

9.
If (X, p) and (Y, q) are two asymmetric normed spaces, the set LC(X, Y) of all continuous linear mappings from (X, p) to (Y, q) is not necessarily a linear space, it is a cone. If X and Y are two Banach lattices and p and q are, respectively, their associated asymmetric norms (p(x) = ‖+‖, q(y) = ‖y +‖), we prove that the positive operators from X to Y are elements of the cone LC(X, Y). We also study the dual space of an asymmetric normed space and finally we give open mapping and closed graph type theorems in the framework of asymmetric normed spaces. The classical results for normed spaces follow as particular cases. The author acknowledges the support of the Ministerio de Educación y Ciencia of Spain and FEDER, under grant MTM2006-14925-C02-01 and Generalitat Valenciana under grant GV/2007/198.  相似文献   

10.
For Tikhonov functionals of the form Ψ(x)=‖Axy Y r +αx X q we investigate a steepest descent method in the dual of the Banach space X. We show convergence rates for the proposed method and present numerical tests.  相似文献   

11.
Several results about convolution and about Fourier coefficients for X-valued functions defined on t he torus satisfying the condition sup ||y||=1∫-π^π|| B (f (e^iθ), y)||dθ/2π〈 ∞ for a bounded bilinear map B : X × Y → Z are presented and some applications are given.  相似文献   

12.
A random variable X is called strongly decomposable into (strong) components Y,Z, if X=Y+Z where Y=φ(X), Z=Xφ(X) are independent nondegenerate random variables and φ is a Borel function. Examples of decomposable and indecomposable random variables are given. It is proved that at least one of the strong components Y and Z of any random variable X is singular. A necessary and sufficient condition is given for a discrete random variable X to be strongly decomposable. Phenomena arising when φ is not Borel are discussed. The Fisher information (on a location parameter) in a strongly decomposable X is necessarily infinite.  相似文献   

13.
We introduce a lower semicontinuous analog, L (X), of the well-studied space of upper semicontinuous set-valued maps with nonempty compact interval images. Because the elements of L (X) contain continuous selections, the space C(X) of real-valued continuous functions on X can be used to establish properties of L (X), such as the two interrelated main theorems. The first of these theorems, the Extension Theorem, is proved in this Part I. The Extension Theorem says that for binormal spaces X and Y, every bimonotone homeomorphism between C(X) and C(Y) can be extended to an ordered homeomorphism between L (X) and L (Y). The second main theorem, the Factorization Theorem, is proved in Part II. The Factorization Theorem says that for binormal spaces X and Y, every ordered homeomorphism between L (X) and L (Y) can be characterized by a unique factorization.  相似文献   

14.
LetE be a 1-injective Banach lattice,X any Banach space andT: E ← X a norm bounded linear operator. Then eitherT is an isomorphism on some copy ofl inE or for all σ > 0 there is φ ∈E + such that ‖Tu‖≦φ (|u|)+σ ‖u‖ for alluE. We deduce the theorem that: A norm order continuous injective Banach lattice is order isomorphic to an (AL)-space.  相似文献   

15.
We study continuous subadditive set-valued maps taking points of a linear space X to convex compact subsets of a linear space Y. The subadditivity means that φ(x 1 + x 2) ⊂ φ(x 1) + φ(x 2). We characterize all pairs of locally convex spaces (X, Y) for which any such map has a linear selection, i.e., there exists a linear operator A: XY such that Axφ(x), xX. The existence of linear selections for a class of subadditive maps generated by differences of a continuous function is proved. This result is applied to the Lipschitz stability problem for linear operators in Banach spaces.  相似文献   

16.
Let A and B be Banach function algebras on compact Hausdorff spaces X and Y and let ‖.‖ X and ‖.‖ Y denote the supremum norms on X and Y, respectively. We first establish a result concerning a surjective map T between particular subsets of the uniform closures of A and B, preserving multiplicatively the norm, i.e. ‖Tf Tg Y = ‖fg X , for certain elements f and g in the domain. Then we show that if α ∈ ℂ {0} and T: AB is a surjective, not necessarily linear, map satisfying ‖fg + α X = ‖Tf Tg + α Y , f,gA, then T is injective and there exist a homeomorphism φ: c(B) → c(A) between the Choquet boundaries of B and A, an invertible element ηB with η(Y) ⊆ {1, −1} and a clopen subset K of c(B) such that for each fA,
$ Tf\left( y \right) = \left\{ \begin{gathered} \eta \left( y \right)f\left( {\phi \left( y \right)} \right) y \in K, \hfill \\ - \frac{\alpha } {{\left| \alpha \right|}}\eta \left( y \right)\overline {f\left( {\phi \left( y \right)} \right)} y \in c\left( B \right)\backslash K \hfill \\ \end{gathered} \right. $ Tf\left( y \right) = \left\{ \begin{gathered} \eta \left( y \right)f\left( {\phi \left( y \right)} \right) y \in K, \hfill \\ - \frac{\alpha } {{\left| \alpha \right|}}\eta \left( y \right)\overline {f\left( {\phi \left( y \right)} \right)} y \in c\left( B \right)\backslash K \hfill \\ \end{gathered} \right.   相似文献   

17.
Suppose that(T t )t>0 is aC 0 semi-group of contractions on a Banach spaceX, such that there exists a vectorxX, ‖x‖=1 verifyingJ −1(Jx)={x}, whereJ is the duality mapping fromX toP(X *). If |<T t x,f>|→1, whent→+∞ for somefX *, ‖f‖≤1 thenx is an eigenvector of the generatorA, associated with a purcly imaginary eigenvalue. Because of Lin's example [L], the hypothesis onxX is the best possible. If the hypothesisJ −1(Jx)={x} is not verified, we can prove that ifJx is a singleton and ifJ −1(Jx) is weakly compact, then if |<T t x, f>|→1, whent→+∞ for somefX *, ‖f‖≤1, there existsyJ −1(Jx) such thaty is an eigenvector of the generatorA, associated with a purely imaginary eigenvalue. We give also a counter-example in the case whereX is one of the spaces ℓ1 orL 1.  相似文献   

18.
It is shown that there is a constantK so that, for every separable metric spaceX, there is a mapT:Xc o satisfyingd(x, y)≦‖Tx−Ty‖≦Kd(x, y) for everyx, y ∈ X. This is a part of the author's Ph.D. Thesis prepared at the Hebrew University of Jerusalem, under the supervision of Professor J. Lindenstrauss.  相似文献   

19.
Let F be a field of characteristic ≠2 and φ be a quadratic form over F. By X φ we denote the projective variety given by the equation φ=0. For each positive even integer d≥8 (except for d=12) we construct a field F and a pair φ, ψ of anisotropic d-dimensional forms over F such that the Chow motives of X φ and X ψ coincide but . For a pair of anisotropic (2 n -1)-dimensional quadrics X and Y, we prove that existence of a rational morphism YX is equivalent to existence of a rational morphism YX. Received: 27 September 1999 / Revised version: 27 December 1999  相似文献   

20.
Let X be a metric space, ε^n(X) be the standard trivial Lip n-bundle over X, and Φ be a Lip automorphism germ of ε^n(X). This paper proves that there is a Lip automorphism Φ‘ of ε^n(X) such that the germ of Φ‘ is Φ.  相似文献   

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