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1.
Let M be a von Neumann algebra with separating and cyclic vector ξ0. The map 0 → x1ξ0 with x?M has a least closed extension S. Tomita proved that the isometric involution J and the positive self-adjoint operator Δ obtained from the polar decomposition S = JΔ12 of S satisfy JMJ = M′ and Δit?it = M for any real t. More generally, he obtained similar results for the left von Neumann algebra of any generalized Hilbert algebra. In this paper a shorter proof of his results is given.  相似文献   

2.
Let B be the open unit ball of Cn, n > 1. Let I (for “inner”) be the set of all u ? H °(B) that have ¦u¦ = 1 a.e. on the boundary S of B. Aleksandrov proved recently that there exist nonconstant u ? I. This paper strengthens his basic theorem and provides further information about I and the algebra Q generated by I. Let XY be the finite linear span of products xy, x ? X, y ? Y, and let ¦X¦ be the norm closure, in L = L(S), of X. Some results: set I is dense in the unit ball of H(B) in the compact-open topology. On S, Q?Q is weak1-dense in L, ¦Q? does not contain H, C(S) ?¦Q?H¦ ≠ ¦H?H¦ ≠ L. (When n = 1, ¦Q¦ = Hand ¦Q?Q¦ = L.) Every unimodular ? ? L is a pointwise limit a.e. of products uv?, u ? I, ν ? I. The zeros of every ? ? 0 in the ball algebra (but not of every H-function) can be matched by those of some u ? I, as can any finite number of derivatives at 0 if ∥?∥ < 1. However, ?u cannot be bounded in B if u ? I is non-constant.  相似文献   

3.
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)|VVnk}, where Vnk={V|V?Rk,|V|=n, d(V)=1}. A set VVnk 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)|VVnk}. It is easily observed that δk(n)=O(n?1k). Hence, ck=lim supn→∞δk(n)n1k exists. It is shown that for all n, ckn?1k≤δk(n), and hence, for all n, lk(n)≥ ckn1?1k. For k=2, this implies that l2(n)≥(π212)14n12, which generalizes an observation of Fejes-Toth that limn→∞l2(n)n?12≥(π212)14. It is also shown that lk(n) ≤ [(3?√3)k(k?1)]nδk(n) + o(n1?1k) ≤ [(3?√3)k(k?1)]n1?1k + o(n1?1k). 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 l2(n)≤(πn2)12 + O(1).  相似文献   

4.
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   

5.
Let A(S) be the sup-normed Banach algebra of analytic functions with continuous boundary values on the compact bordered Riemann surface S.For (?) in A(S)?1exp(A(S)), the colength of (?) is defined by ∥(?)∥ = 12log inf{∥ g ∥ ∥ g?1 ∥; g ? (?)}. Colength is shown to induce a norm on the cohomology group H1(S,R) dual to the norm induced on the homology group H1(S,R) by harmonic length, or, equivalently, dual to the norm on Re A(S).The existence and uniqueness of extremal functions for the colength functional is demonstrated. The aforementioned norms are shown to determine the conformal structure of S (up to reflection) and to be related to the mapping properties of S.  相似文献   

6.
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 (i ? j, H \?bo2 K, E \?boαF) 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 H \?bo2 K into E \?boαF. Then we exhibit an example which shows that the answer cannot be positive in full generality. Finally we prove that if F=Lp(X,X,λ) for some σ-finite measure λ ? 0 then (i?j, H?2K,Lp(X,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,X,λ)).  相似文献   

7.
Let PT denote the orthogonal projection of L2(R1, ) onto the space of entire functions of exponential type ? T which are square summable on the line with respect to the measure dΔ(γ) = ¦ h(γ)¦2, and let G denote the operator of multiplication by a suitably restricted complex valued function g. It is shown that if 2 + 1)?1log ¦ h(γ)¦ is summable, if ¦ h ¦?2 is locally summable, and if hh# belongs to the span in L of e?iyTH:T ? 0, in which h is chosen to be an outer function and h#(γ) agrees with the complex conjugate of h(γ) on the line, then
lim traceT↑∞{(PTGPT)n ? PTGnPT}
exists and is independent of h for every positive integer n. This extends the range of validity of a formula due to Mark Kac who evaluated this limit in the special case h = 1 using a different formalism. It also extends earlier results of the author which were established under more stringent conditions on h. The conclusions are based in part upon a preliminary study of a more general class of projections.  相似文献   

8.
Let θ(n) denote the maximum likelihood estimator of a vector parameter, based on an i.i.d. sample of size n. The class of estimators θ(n) + n?1q(θ(n)), with q running through a class of sufficiently smooth functions, is essentially complete in the following sense: For any estimator T(n) there exists q such that the risk of θ(n) + n?1q(θ(n)) exceeds the risk of T(n) by an amount of order o(n?1) at most, simultaneously for all loss functions which are bounded, symmetric, and neg-unimodal. If q1 is chosen such that θ(n) + n?1 q1(n)) is unbiased up to o(n?12), then this estimator minimizes the risk up to an amount of order o(n?1) in the class of all estimators which are unbiased up to o(n?12).The results are obtained under the assumption that T(n) admits a stochastic expansion, and that either the distributions have—roughly speaking—densities with respect to the lebesgue measure, or the loss functions are sufficiently smooth.  相似文献   

9.
If f is a positive function on (0, ∞) which is monotone of order n for every n in the sense of Löwner and if Φ1 and Φ2 are concave maps among positive definite matrices, then the following map involving tensor products:
(A,B)?f[Φ1(A)?12(B)]·(Φ1(A)?I)
is proved to be concave. If Φ1 is affine, it is proved without use of positivity that the map
(A,B)?f[Φ1(A)?Φ2(B)?1]·(Φ1(A)?I)
is convex. These yield the concavity of the map
(A,B)?A1?p?Bp
(0<p?1) (Lieb's theorem) and the convexity of the map
(A,B)?A1+p?B?p
(0<p?1), as well as the convexity of the map
(A,B)?(A·log[A])?I?A?log[B]
.These concavity and convexity theorems are then applied to obtain unusual estimates, from above and below, for Hadamard products of positive definite matrices.  相似文献   

10.
Let (L2)B?? and (L2)b?? be the spaces of generalized Brownian functionals of the white noises ? and ?, respectively. A Fourier transform from (L2)B?? into (L2)b?? is defined by ??(?) = ∫S1: exp[?i ∫R?(t) ?(t) dt]: b??(B?) dμ(B?), where : :b? denotes the renormalization with respect to ? and μ is the standard Gaussian measure on the space S1 of tempered distributions. It is proved that the Fourier transform carries ?(t)-differentiation into multiplication by i?(t). The integral representation and the action of?? as a generalized Brownian functional are obtained. Some examples of Fourier transform are given.  相似文献   

11.
Let B be a body in R3 and let S denote the boundary of B. The surface S is described by S = {(x, y, z): (x2 + y2)12 = f(z), ?1 ? z ? 1}, where f is an analytic function that is real and positive on (?1, 1) and f(±1) = 0. An algorithm is described for computing the scattered field due to a plane wave incident field, under Leontovich boundary conditions. The Galerkin method of solution used here leads to a block diagonal matrix involving 2M + 1 blocks, each block being of order 2(2N + 1). If, e.g., N = O(M2), the computed scattered field is accurate to within an error bounded by Ce?cN12, where C and c are positive constants depending only on f.  相似文献   

12.
Let A, B be two matrices of the same order. We write A>B(A>?B) iff A? B is a positive (semi-) definite hermitian matrix. In this paper the well-known result if
A>B>θ, then B?1>A?1> θ
(cf. Bellman [1, p.59]) is extended to the generalized inverses of certain types of pairs of singular matrices A,B?θ, where θ denotes the zero matrix of appropriate order.  相似文献   

13.
Let O = limnZ/pnZ, let A = O[g2, g3]Δ, where g2 and g3 are coefficients of the elliptic curve: Y2 = 4X3 ? g2X ? g3 over a finite field and Δ = g23 ? 27g32 and let B = A[X, Y](Y2 ? 4X3 + g2X + g3). Then the p-adic cohomology theory will be applied to compute explicitly the zeta matrices of the elliptic curves, induced by the pth power map on the free A2?ZQ-module H1(X, A2?ZQ). Main results are; Theorem 1.1: X2dY and YdX are basis elements for H1(X, ΓA1(X)2?ZQ); Theorem 1.2: YdX, X2dY, Y?1dX, Y?2dX and XY?2dX are basis elements for H1(X ? (Y = 0), ΓA1(X)2?ZQ), where X is a lifting of X, and all the necessary recursive formulas for this explicit computation are given.  相似文献   

14.
Let etSande?tT be (C0)-semigroups on a Banach space X. Their tensor product L(t) is defined by L(t)A = etSAetT (A?B(X)) and has the generator Δ formally of the form ΔA = SA ? AT. Under the assumption that {L(t); t ? 0} is bounded, we investigate the Abel limit and the Cesàro limit of L(t)A at ∞. If gWsu] denotes the set of operators A for which the Abel limit Ps(A) [resp. Pu(A)] exists in the strong [resp. uniform] operator topology, then
N(Δ)⊕R(Δ) = ωu ? ωs ? N(Δ) + R(Δ)
and the limit defines a projection Ps[Pu] from Ωs [resp. Ωu] onto N(Δ) with N(Δ) with R(Δ) = N(Pu) ? N(Pu) ? R(Δ). If, in addition, S and T are Hilbert space normal operators such that gq(S) ∩ gq(T) ≠ φ, then Ωu contains all compact operators.  相似文献   

15.
On a compact Kähler manifold of complex dimension m ? 2, let us consider the change of Kähler metric g′λ\?gm = gλ\?gm + ?λ\?gmφ. Let F?C(V × R) be a function everywhere > 0 and v a real number ≠ 0. When 0 < C?1 ? F(x, t) ? C(¦t¦a + 1) for all (x, t) ?V × ] ?∞, t0], where C and t0 are constants and 1 ? a < m(m ? 1), one exhibits a function φ?C (V) such that ¦g′∥g¦?1 = eν\?gfF(x, φ ? \?gf) (¦g¦ and ¦g′¦ the determinants of the metrics g and g′, \?gf = (mes V)?1 ∝ φ dV).  相似文献   

16.
Let SφP1 be an elliptic fibration on a K3 surface S. Then the composition S[n]πS(n)symnφPn gives an Abelian fibration on S[n]. Let E be the exceptional divisor of π, then symnφ°π(E) is of dimension n?1. We prove the inverse in this Note. To cite this article: B. Fu, C. R. Acad. Sci. Paris, Ser. I 337 (2003).  相似文献   

17.
If Ω is a weakly pseudoconvex domain in a Stein manifold, then the spectrum of the Frechet Algebra A(\?gW) is isomorphic to \?gW, A(\?gW) denotes the space of holomorphic functions smooth up to the boundary. The spectrum of the uniform algebra A(\?gW) is also isomorphic to -Ω. As a corollary we prove an approximation theorem for plurisubharmonic functions in Ω continuous in -Ω.  相似文献   

18.
Let N denote a connected, simply connected nilpotent Lie group with discrete cocompact subgroup Γ. Let U denote the quasi-regular representation on N on L2(NΓ). L2(NΓ) can be written as a direct sum of primary subspaces with respect to U. A realization for the projections of L2(NΓ)) onto these primary summands is given in this paper.  相似文献   

19.
20.
On Rn, n?1 and n≠2, we prove the existence of a sharp constant for Sobolev inequalities with higher fractional derivatives. Let s be a positive real number. For n>2s and q=2nn?2s any function f∈Hs(Rn) satisfies
6f62q?Sn,s(?Δ)s/2f22,
where the operator (?Δ)s in Fourier spaces is defined by (?Δ)sf(k):=(2π|k|)2sf(k). To cite this article: A. Cotsiolis, N.C. Tavoularis, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 801–804.  相似文献   

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