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Let G be an additive subgroup of a normed space X. We say that a point is weakly separated (resp. -separated) from G if it can be separated from G by a continuous character (resp. by a continuous positive definite function). Let T : X → Y be a continuous linear operator. Consider the following conditions:
(ws) if , then x is weakly separated from G;
(ps) if , then x is -separated from G;
(wp) if Tx is -separated from T(G), then x is weakly separated from G.
By (resp. , ) we denote the class of operators T : X → Y which satisfy (ws) (resp. (ps), (wp)) for all and all subgroups G of X. The paper is an attempt to describe the above classes of operators for various Banach spaces X, Y. It is proved that if X, Y are Hilbert spaces, then is the class of Hilbert-Schmidt operators. It is also shown that if T is a Hilbert-to-Banach space operator with finite ℓ-norm, then .
相似文献
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W. Banaszczyk 《Mathematische Annalen》1993,296(1):625-635
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Let G be the group "ax + b" of affine transformations of the line and let U be a neighbourhood of 1 in G. It is proved that there is another neighborhood V of 1 such that to each finite sequence g1,...,gn V there corresponds a sequence of signs 1,...,n = ±1 with
U for k = 1,...,n. This implies that G satisfies the following analogue of the Dvoretzky-Hanani theorem: to each sequence
converging to 1 in G there corresponds a sequence of signs k = ±1 such that the infinite product
is convergent. 相似文献
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Wojciech Banaszczyk 《Random Structures and Algorithms》1998,12(4):351-360
Let ‖·‖ be the Euclidean norm on R n and γn the (standard) Gaussian measure on R n with density (2π)−n/2e. It is proved that there is a numerical constant c>0 with the following property: if K is an arbitrary convex body in R n with γn(K)≥1/2, then to each sequence u1,…,um∈ R n with ‖u1‖,…,‖um‖≤c there correspond signs ε1,…,εm=±1 such that ∑mi=1εiui∈K. This improves the well-known result obtained by Spencer [Trans. Amer. Math. Soc. 289 , 679–705 (1985)] for the n-dimensional cube. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 12: 351–360, 1998 相似文献
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Wojciech Banaszczyk 《Random Structures and Algorithms》2012,40(3):301-316
Let x1,…,xm∈ \input amssym $ \Bbb R$ n be a sequence of vectors with ∥xi∥2 ≤ 1 for all i. It is proved that there are signs ε1,…,εm = ±1 such that where C1, C2 are some numerical constants. It is also proved that there are signs ε,…,ε = ±1 and a permutation π of {1,…,m} such that where C′ is some other numerical constant. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011 相似文献
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W. Banaszczyk 《Discrete and Computational Geometry》1995,13(1):217-231
LetL be a lattice and letU be ano-symmetric convex body inR
n
. The Minkowski functional ∥ ∥
U
ofU, the polar bodyU
0, the dual latticeL
*, the covering radius μ(L, U), and the successive minima λ
i
(L,U)i=1,...,n, are defined in the usual way. Let ℒ
n
be the family of all lattices inR
n
. Given a pairU,V of convex bodies, we define
and kh(U, V) is defined as the smallest positive numbers for which, given arbitraryL∈ℒ
n
andu∈R
n
/(L+U), somev∈L
* with ∥v∥
V
≤sd(uv, ℤ) can be found. Upper bounds for jh(U, U
0), j=k, l, m, belong to the so-called transference theorems in the geometry of numbers. The technique of Gaussian-like measures
on lattices, developed in an earlier paper [4] for euclidean balls, is applied to obtain upper bounds for jh(U, V) in the case whenU, V aren-dimensional ellipsoids, rectangular parallelepipeds, or unit balls inl
p
n
, 1≤p≤∞. The gaps between the upper bounds obtained and the known lower bounds are, roughly speaking, of order at most logn asn→∞. It is also proved that ifU is symmetric through each of the coordinate hyperplanes, then jh(U, U
0) are less thanCn logn for some numerical constantC. 相似文献