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1.
We introduce the class of operators on Banach spaces having property (H) and study Weyl’s theorems, and related results for operators which satisfy this property. We show that a- Weyl’s theorem holds for every decomposable operator having property (H). We also show that a-Weyl’s theorem holds for every multiplier T of a commutative semi-simple regular Tauberian Banach algebra. In particular every convolution operator Tμ of a group algebra L1(G), G a locally compact abelian group, satisfies a-Weyl’s theorem. Similar results are given for multipliers of other important commutative Banach algebras. 相似文献
2.
Nick Dungey 《Probability Theory and Related Fields》2007,137(3-4):429-442
Let G be a compactly generated, locally compact group, and let T be the operator of convolution with a probability measure μ on G. Our main results give sufficient conditions on μ for the operator T to be analytic in L
p
(G), 1 < p < ∞, where analyticity means that one has an estimate of form for all n = 1, 2, ... in L
p
operator norm. Counterexamples show that analyticity may not hold if some of the conditions are not satisfied. 相似文献
3.
An error analysis of Runge–Kutta convolution quadrature is presented for a class of non-sectorial operators whose Laplace
transform satisfies, besides the standard assumptions of analyticity in a half-plane Re s > σ
0 and a polynomial bound
\operatornameO(|s|m1){\operatorname{O}(|s|^{\mu_1})} there, the stronger polynomial bound
\operatornameO(sm2){\operatorname{O}(s^{\mu_2})} in convex sectors of the form
|\operatorname*arg s| £ p/2-q{|\operatorname*{arg} s| \leq \pi/2-\theta} for θ > 0. The order of convergence of the Runge–Kutta convolution quadrature is determined by μ
2 and the underlying Runge–Kutta method, but is independent of μ
1. Time domain boundary integral operators for wave propagation problems have Laplace transforms that satisfy bounds of the
above type. Numerical examples from acoustic scattering show that the theory describes accurately the convergence behaviour
of Runge–Kutta convolution quadrature for this class of applications. Our results show in particular that the full classical
order of the Runge–Kutta method is attained away from the scattering boundary. 相似文献
4.
A direction–length framework is a pair (G,p) where G=(V;D,L) is a ‘mixed’ graph whose edges are labelled as ‘direction’ or ‘length’ edges and p is a map from V to ℝ
d
for some d. The label of an edge uv represents a direction or length constraint between p(u) and p(v). Let G
+ be obtained from G by adding, for each length edge e of G, a direction edge with the same end vertices as e. We show that (G,p) is bounded if and only if (G
+,p) is infinitesimally rigid. This gives a characterization of when (G,p) is bounded in terms of the rank of the rigidity matrix of (G
+,p). We use this to characterize when a mixed graph is generically bounded in ℝ
d
. As an application we deduce that if (G,p) is a globally rigid generic framework with at least two length edges and e is a length edge of G then (G∖e,p) is bounded. 相似文献
5.
A Banach space X is said to be an extremely non-complex space if the norm equality ∥Id +T
2∥ = 1+∥T
2∥ holds for every bounded linear operator T on X. We show that every extremely non-complex Banach space has positive numerical index, it does not have an unconditional basis
and that the infimum of diameters of the slices of its unit ball is positive. 相似文献
6.
A linear and bounded operator T between Banach spaces X and Y has Fourier type 2 with respect to a locally compact abelian group G if there exists a constant c > 0 such that∥T ∥2 ≤ c∥f∥2 holds for all X‐valued functions f ∈ LX2(G) where is the Fourier transform of f. We show that any Fourier type 2 operator with respect to the classical groups has Fourier type 2 with respect to any locally compact abelian group. This generalizes previous special results for the Cantor group and for closed subgroups of ?n. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
7.
It is classical that amongst all spaces Lp (G), 1 ≤ p ≤ ∞, for , or say, only L2 (G) (that is, p = 2) has the property that every bounded Borel function on the dual group Γ determines a bounded Fourier multiplier operator
in L2 (G). Stone’s theorem asserts that there exists a regular, projection-valued measure (of operators on L2 (G)), defined on the Borel sets of Γ, with Fourier-Stieltjes transform equal to the group of translation operators on L2 (G); this fails for every p ≠ 2. We show that this special status of L2 (G) amongst the spaces Lp (G), 1 ≤ p ≤ ∞, is actually more widespread; it continues to hold in a much larger class of Banach function spaces defined over G (relative to Haar measure).
相似文献
8.
S. J. Bhatt 《Proceedings Mathematical Sciences》1998,108(3):283-303
A completely positive operator valued linear map ϕ on a (not necessarily unital) Banach *-algebra with continuous involution
admits minimal Stinespring dilation iff for some scalark > 0, ϕ(x)*ϕ(x) ≤ kϕ(x*x) for allx iff ϕ is hermitian and satisfies Kadison’s Schwarz inequality ϕ(h)
2
≤ kϕ(h
2) for all hermitianh iff ϕ extends as a completely positive map on the unitizationA
e
of A. A similar result holds for positive linear maps. These provide operator state analogues of the corresponding well-known
results for representable positive functionals. Further, they are used to discuss (a) automatic Stinespring representability
in Banach *-algebras, (b) operator valued analogue of Bochner-Weil-Raikov integral representation theorem, (c) operator valued
analogue of the classical Bochner theorem in locally compact abelian groupG, and (d) extendability of completely positive maps from *-subalgebras. Evans’ result on Stinespring respresentability in
the presence of bounded approximate identity (BAI) is deduced. A number of examples of Banach *-algebras without BAI are discussed
to illustrate above results. 相似文献
9.
Wojciech Jaworski 《Journal d'Analyse Mathématique》1996,68(1):183-208
Given a probability measure μ on a locally compact second countable groupG the space of bounded μ-harmonic functions can be identified withL
∞(η, α) where (η, α) is a BorelG-space with a σ-finite quasiinvariant measure α. Our goal is to show that when μ is an arbitrary spread out probability measure
on a connected solvable Lie groupG then the μ-boundary (η, α) is a contractive homogeneous space ofG. Our approach is based on a study of a class of strongly approximately transitive (SAT) actions ofG. A BorelG-space η with a σ-finite quasiinvariant measure α is called SAT if it admits a probability measurev≪α, such that for every Borel set A with α(A)≠0 and every ε>0 there existsg∈G with ν(gA)>1−ε. Every μ-boundary is a standard SATG-space. We show that for a connected solvable Lie group every standard SATG-space is transitive, characterize subgroupsH⊆G such that the homogeneous spaceG/H is SAT, and establish that the following conditions are equivalent forG/H: (a)G/H is SAT; (b)G/H is contractive; (c)G/H is an equivariant image of a μ-boundary. 相似文献
10.
S. Shkarin 《Integral Equations and Operator Theory》2009,64(1):115-136
It is shown that if 1 < p < ∞ and X is a subspace or a quotient of an ℓp-direct sum of finite dimensional Banach spaces, then for any compact operator T on X such that ∥I + T∥ > 1, the operator I + T attains its norm. A reflexive Banach space X and a bounded rank one operator T on X are constructed such that ∥I + T∥ > 1 and I + T does not attain its norm.
The author would like to thank E. Shargorodsky for his interest and comments. 相似文献
11.
We define Hilbert transform and conjugate Poisson integrals associated with the Jacobi differential operator on (0, +∞). We
prove that these operators are bounded in the appropriate Lebesgue spaces L
p
, 1 < p < +∞. In this study, the tools used are the Littlewood–Paley g-functions associated with the Poisson semigroup and the supplementary Poisson semigroup which we introduce in this paper. 相似文献
12.
Wojciech Jaworski 《Journal d'Analyse Mathématique》1998,74(1):235-273
We prove that, given an arbitrary spread out probability measure μ on an almost connected locally compact second countable
groupG, there exists a homogeneous spaceG/H, called the μ-boundary, such that the space of bounded μ-harmonic functions can be identified withL
∞
(G/H). The μ-boundary is an amenable contractive homogeneous space. We also establish that the canonical projection onto the μ-boundary
of the right random walk of law μ always converges in probability and, whenG is amenable, it converges almost surely. The μ-boundary can be characterised as the largest homogeneous space among those
homogeneous spaces in which the canonical projection of the random walk converges in probability. 相似文献
13.
Suppose {G1(t)}t ≥ 0 and {G2(t)t ≥ 0 be two semigroups on an infinite dimensional separable reflexive Banach space X. In this paper we give sufficient conditions for tensor product semigroup G(t): X → G2(t)X G1(t) to become chaotic in L with the strong operator topology and chaotic in the ideal of compact operators on X with the norm operator topology. 相似文献
14.
D. S. Anisimov 《Journal of Mathematical Sciences》2006,139(2):6363-6368
A version of Grothendieck’s inequality says that any bounded linear operator acting from a Banach lattice X to a Banach lattice
Y acts from X(ℓ2) to Y (ℓ2) as well. A similar statement is proved for Hardy-type subspaces in lattices of measurable functions. Namely, let X be a
Banach lattice of measurable functions on the circle, and let an operator T act from the corresponding subspace of analytic
functions XA to a Banach lattice Y or, if Y is also a lattice of measurable functions on the circle, to the quotient space Y/YA. Under certain mild conditions on the lattices involved, it is proved that T induces an operator acting from XA(ℓ2) to Y (ℓ2) or to Y/YA(ℓ2), respectively. Bibliography: 7 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 327, 2005, pp. 5–16. 相似文献
15.
The purpose of this note is to describe some algebraic conditions on a Banach algebra which force it to be finite dimensional.
One of the main results in Theorem 2 which states that for a locally compact groupG, G is compact if there exists a measure μ in Soc(L
1(G)) such that μ(G) ≠ 0. We also prove thatG is finite if Soc(M(G)) is closed and every nonzero left ideal inM(G) contains a minimal left ideal. 相似文献
16.
Alfred Geroldinger David J. Grynkiewicz Wolfgang A. Schmid 《Acta Mathematica Hungarica》2011,131(4):323-345
For a finite abelian group G and a positive integer d, let s
dℕ(G) denote the smallest integer ℓ∈ℕ0 such that every sequence S over G of length |S|≧ℓ has a nonempty zero-sum subsequence T of length |T|≡0 mod d. We determine s
dℕ(G) for all d≧1 when G has rank at most two and, under mild conditions on d, also obtain precise values in the case of p-groups. In the same spirit, we obtain new upper bounds for the Erdős–Ginzburg–Ziv constant provided that, for the p-subgroups G
p
of G, the Davenport constant D(G
p
) is bounded above by 2exp (G
p
)−1. This generalizes former results for groups of rank two. 相似文献
17.
Riddhi Shah 《Israel Journal of Mathematics》1999,110(1):189-218
It is shown that the limit μ of a commutative infinitesimal triangular system Δ on a totally disconnected locally compact
groupG is embeddable in a continuous one-parameter convolution semigroup if either (1)G is a compact extension of a closed solvable normal subgroup or (2)G is discrete and Δ is normal or (3)G is a discrete linear group over a field of characteristic zero. For a special triangular system of convolution powers
, the above is shown to hold without any of the conditions (1)–(3). For a general locally compact groupG necessary conditions are obtained for the embeddability of a shift of limit μ of Δ; in particular, the conditions are trivially
satisfied whenG is abelian. Also, the embedding of a limit of a symmetric system onG is shown to hold under condition (1) as above. 相似文献
18.
19.
Given a locally compact group G acting on a locally compact space X and a probability measure σ on G, a real Borel function f on X is called σ-harmonic if it satisfies the convolution equation . We give conditions for the absence of nonconstant bounded harmonic functions. We show that, if G is a union of σ-admissible neighbourhoods of the identity, relative to X, then every bounded σ-harmonic function on X is constant. Consequently, for spread out σ, the bounded σ-harmonic functions are constant on each connected component of a [SIN]-group and, if G acts strictly transitively on a splittable metric space X, then the bounded σ-harmonic functions on X are constant which extends Furstenberg’s result for connected semisimple Lie groups. 相似文献
20.
Given a locally compact group G acting on a locally compact space X and a probability measure σ on G, a real Borel function f on X is called σ-harmonic if it satisfies the convolution equation . We give conditions for the absence of nonconstant bounded harmonic functions. We show that, if G is a union of σ-admissible neighbourhoods of the identity, relative to X, then every bounded σ-harmonic function on X is constant. Consequently, for spread out σ, the bounded σ-harmonic functions are constant on each connected component of
a [SIN]-group and, if G acts strictly transitively on a splittable metric space X, then the bounded σ-harmonic functions on X are constant which extends Furstenberg’s result for connected semisimple Lie groups.
(Received 13 June 1998; in revised form 31 March 1999) 相似文献