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1.
LetA
e
be the algebra obtained by adjoining identity to a non-unital Banach algebra (A, ∥ · ∥). Unlike the case for aC*-norm on a Banach *-algebra,A
e
admits exactly one uniform norm (not necessarily complete) if so doesA. This is used to show that the spectral extension property carries over fromA to A
e
. Norms onA
e
that extend the given complete norm ∥ · ∥ onA are investigated. The operator seminorm ∥ · ∥op onA
e
defined by ∥ · ∥ is a norm (resp. a complete norm) iffA has trivial left annihilator (resp. ∥ · ∥op restricted toA is equivalent to ∥ · ∥). 相似文献
2.
Let K be a complete ultrametric algebraically closed field and let A be the K-Banach algebra of bounded analytic functions in the disk D: |x| < 1. Let Mult(A, ∥ · ∥) be the set of continuous multiplicative semi-norms of A, let Mult
m
(A, ∥ · ∥) be the subset of the ϕ ∈ Mult(A, ∥ · ∥) whose kernel is a maximal ideal and let Mult
a
(A, ∥ · ∥) be the subset of the ϕ ∈ Mult
m
(A, ∥ · ∥) whose kernel is of the form (x − a)A, a ∈ D ( if ϕ ∈ Mult
m
(A, ∥ · ∥) \ Mult
a
(A, ∥ · ∥), the kernel of ϕ is then of infinite codimension). We examine whether Mult
a
(A, ∥ · ∥) is dense inside Mult
m
(A, ∥ · ∥) with respect to the topology of simple convergence. This a first step to the conjecture of density of Mult
a
(A, ∥ · ∥) in the whole set Mult(A, ∥ · ∥): this is the corresponding problem to the well-known complex corona problem. We notice that if ϕ ∈ Mult
m
(A, ∥ · ∥) is defined by an ultrafilter on D, then ϕ lies in the closure of Mult
a
(A, ∥ · ∥). Particularly, we show that this is case when a maximal ideal is the kernel of a unique ϕ ∈ Multm(A, ∥ · ∥). Particularly, when K is strongly valued all maximal ideals enjoy this property. And we can prove this is also true when K is spherically complete, thanks to the ultrametric holomorphic functional calculus. More generally, we show that if ψ ∈ Mult(A, ∥ · ∥) does not define the Gauss norm on polynomials (∥ · ∥), then it is defined by a circular filter, like on rational
functions and analytic elements. As a consequence, if ψ ∈
Multm(A, ∥ · ∥) \ Multa(A, ∥ · ∥) or if φ does not lie in the closure of Mult
a
(A, ∥ · ∥), then its restriction to polynomials is the Gauss norm. The first situation does happen. The second is unlikely.
The text was submitted by the authors in English. 相似文献
3.
Résumé. Soit A une algèbre réelle sans diviseurs de zéro. On suppose que l’espace vectoriel A est muni d’une norme ∥.∥ préhilbertienne vérifiant ∥a
2∥ ≤ ∥a∥2 pour tout . Alors A est de dimension finie dans chacun des quatre cas suivants :
A est isomorphe à ou dans les deux premiers cas et isomorphe à ou dans les deux derniers cas.
1. | A est commutative contenant un élément non nul a tel que ∥ax∥ = ∥a∥ ∥x∥ pour tout , |
2. | A est commutative algébrique et ∥a 2∥ = ∥a∥2 pour tout , |
3. | A est alternative contenant un élément unité e tel que ∥e∥ = 1, |
4. | A est alternative contenant un élément central non nul a tel que ∥ax∥ = ∥a∥ ∥x∥ pour tout . |
Let A be a real algebra without divisor of zero. Assuming that a vector space A is endowed with a pre-Hilbert norm ∥.∥ satisfying ∥a 2∥ ≤ ∥a∥2 for all . Then A is finite dimensional in the four following cases :A is isomorphic to or in the two first cases and isomorphic to or in the two last cases.
1. A is a commutative containing a nonzero element a such that ∥ax∥ = ∥a∥∥x∥ for all , 2. A is a commutative algebraic and ∥a 2∥ = ∥a∥2 for all , 3. A is an alternative containing a unit element e such that ∥e∥ = 1, 4. A is an alternative containing a nonzero central element a such that ∥ax∥ = ∥ a∥∥x∥ for all .
相似文献
4.
Szilvia Szilágyi 《Order》2008,25(4):321-333
We present a characterization of the maximal compatible extensions of a given compatible partial order ≤
r
on a unary algebra (A,f ). These extensions can be constructed by using the compatible linear extensions of ≤
r*, where (A*,f*) is the so called contracted quotient algebra of (A,f) and the compatible partial order ≤
r* on (A*,f*) is naturally induced by ≤
r
. Using this characterization, we determine the intersection of the maximal compatible extensions of ≤
r
.
相似文献
5.
Given a specification linear operatorS, we want to test an implementation linear operatorA and determine whether it conforms to the specification operator according to an error criterion. In an earlier paper [3],
we studied a worst case error in which we test whether the error is no more than a given bound ε>0 for all elements in a given
setF, i.e., sup
fεf∥Sf—Af∥≤ε. In this work, we study the average error instead, i. e., ∫
F
∥Sf-Af∥2μ(df)ɛ≤2, where μ is a probability measure onF. We assume that an upper boundK on the norm of the difference ofS andA is given a priori. It turns out that any finite number of tests is in general inconclusive with the average error. Therefore,
as in the worst case, we allow a relaxation parameter α>0 and test for weak conformance with an error bound (1+α)ε. Then a
finite number of tests from an arbitrary orthogonal complete sequence is conclusive. Furthermore, the eigenvectors of the
covariance operatorC
μ of the probability measure μ provide an almost optimal test sequence. This implies that the test set isuniversal; it only depends on the set of valid inputsF and the measure μ, and is independent ofS, A, and the other parameters of the problem. However, the minimal number of tests does depend on all the parameters of the testing
problem, i.e., ε, α,K, and the eigenvalues ofC
μ. In contrast to the worst case setting, it also depends on the dimensiond of the range space ofS andA.
This work was done while consulting at Bell Laboratories, and is partially supported by the National Science Foundation and
the Air Force Office of Scientific Research. 相似文献
6.
A. Moutassim 《Advances in Applied Clifford Algebras》2008,18(2):255-267
Résumé. Soit A une algèbre réelle. On suppose que l’espace vectoriel A est muni d’une norme ∥.∥ préhilbertienne vérifiant ∥a
2∥ = ∥a∥2 pour tout . Si A est flexible, sans diviseurs de zéro et de dimension ≤ 4, alors A est isomorphe à ou , ce qui généralise un théorème d’El-Mallah [1]. Si A est flexible, sans diviseurs de zéro, contenant un idempotent central et vérifiant la propriété d’Osborn, alors A est de dimension finie et isomorphe à , ou . Enfin nous montrons qu’une algèbre normée préhilbertienne unitaire d’unité e telle que ∥e∥ = 1 est flexible et vérifie ∥a
2∥ = ∥ a∥2.
Let A be a real algebra. Assuming that a vector space A is endowed with a pre-Hilbert norm ∥.∥ satisfying ∥a 2∥ = ∥a∥2 for all . If A is flexible, without divisor of zero and of a dimension ≤ 4, then A is isomorphic to or , which generalize El-Mallah’s theorem [1]. If A is flexible, without divisor of zero, containing a central idempotent and satisfying Osborn’s properties, then A is finite dimensional and isomorphic to , or . Finally we prove that a normed pre-Hilbert algebra with unit e such that ∥e∥ = 1 is flexible and satisfies ∥a 2∥ = ∥a∥2.
相似文献
7.
Von Neumann-Jordan Constants of Absolute Normalized Norms on C^n 总被引:1,自引:0,他引:1
In this note, we give some estimations of the Von Neumann-Jordan constant C
N J
(∥·∥ψ) of Banach space (ℂ
n
, ∥·∥ψ), where ∥·∥ψ is the absolute normalized norm on ℂ
n
given by function ψ. In the case where ψ and φ are comparable, n=2 and C
N J
(∥·∥ψ)=1, we obtain a formula of computing C
N J
(∥·∥ψ). Our results generalize some results due to Saito and others.
Received May 11, 2002, Accepted November 20, 2002
This work is partly supported by NNSF of China (No. 19771056) 相似文献
8.
Tao Xiangxing 《分析论及其应用》1996,12(2):13-19
Let μ be a measure on the upper half-space R
+
n+1
, and v a weight onR
n, we give a characterization for the pair (v, μ) such that ∥M(fv)∥L
Θ
(μ) ⩽ c ∥f∥L
Θ
(μ), where Φ is an N-function satisfying Δ2 condition andMf(x,t), is the maximal function onR
+
n+1
, which was introduced by Ruiz, F. and Torrea, J..
Supported by NSFC. 相似文献
9.
Levent Tunçel 《Mathematical Programming》1999,86(1):219-223
Given an m×n integer matrix A of full row rank, we consider the problem of computing the maximum of ∥B
-1
A∥2 where B varies over all bases of A. This quantity appears in various places in the mathematical programming literature. More recently, logarithm of this number
was the determining factor in the complexity bound of Vavasis and Ye’s primal-dual interior-point algorithm. We prove that
the problem of approximating this maximum norm, even within an exponential (in the dimension of A) factor, is NP-hard. Our proof is based on a closely related result of L. Khachiyan [1].
Received November 13, 1998 / Revised version received January 20, 1999? Published online May 12, 1999 相似文献
10.
Let L be the infinitesimal generator of an analytic semigroup on L2 (Rn) with suitable upper bounds on its heat kernels. Assume that L has a bounded holomorphic functional calculus on L2(Rn). In this paper,we define the Littlewood- Paley g function associated with L on Rn × Rn, denoted by GL(f)(x1, x2), and decomposition, we prove that ‖SL(f)‖p ≈‖GL(f)‖p ≈‖f‖p for 1 < p <∞. 相似文献
11.
Ana Bela CRUZEIRO Xi Cheng ZHANG 《数学学报(英文版)》2006,22(1):101-104
For 1 〈 p ≤2, an L^p-gradient estimate for a symmetric Markov semigroup is derived in a general framework, i.e. ‖Γ^/2(Ttf)‖p≤Cp/√t‖p, where F is a carre du champ operator. As a simple application we prove that F1/2((I- L) ^-α) is a bounded operator from L^p to L^v provided that 1 〈 p 〈 2 and 1/2〈α〈1. For any 1 〈 p 〈 2, q 〉 2 and 1/2 〈α 〈 1, there exist two positive constants cq,α,Cp,α such that ‖Df‖p≤ Cp,α‖(I - L)^αf‖p,Cq,α(I-L)^(1-α)‖Df‖q+‖f‖q, where D is the Malliavin gradient ([2]) and L the Ornstein-Uhlenbeck operator. 相似文献
12.
Based on a novel point of view on 1-dimensional Gaussian quadrature, we present a new approach to d-dimensional cubature formulae. It is well known that the nodes of 1-dimensional Gaussian quadrature can be computed as eigenvalues
of the so-called Jacobi matrix. The d-dimensional analog is that cubature nodes can be obtained from the eigenvalues of certain mutually commuting matrices. These
are obtained by extending (adding rows and columns to) certain noncommuting matrices A1,...,Ad, related to the coordinate operators x1,...,xd, in Rd. We prove a correspondence between cubature formulae and “commuting extensions” of A1,...,Ad, satisfying a compatibility condition which, in appropriate coordinates, constrains certain blocks in the extended matrices
to be zero. Thus, the problem of finding cubature formulae can be transformed to the problem of computing (and then simultaneously
diagonalizing) commuting extensions. We give a general discussion of existence and of the expected size of commuting extensions
and briefly describe our attempts at computing them. 相似文献
13.
R. Zarouf 《Journal of Mathematical Sciences》2009,156(5):819-823
The condition numbers CN(T) = ∥T∥ · ∥T−1∥ of Toeplitz and analytic n × n matrices T are studied. It is shown that the supremum of CN(T) over all such matrices with
∥T∥ ≤ 1 and the given minimum of eigenvalues r = min |λi| > 0 behaves as the corresponding supremum over all n × n matrices (i.e., as (Kronecker)), and this equivalence is uniform in n and r. The proof is based on a use of the Sarason-Sz.Nagy-Foias commutant
lifting theorem. Bibliography: 2 titles.
Published in Zapiski Nauchnykh Seminarov POMI, Vol. 355, 2008, pp. 173–179. 相似文献
14.
We determine the best approximation of an arbitrary power A
k
of an unbounded self-adjoint operator A in a Hilbert space H on the class {x ∈ D(A
r
) : ∥A
r
x∥ ≤ 1}, k < r. 相似文献
15.
M. T. Tarashchanskii 《Ukrainian Mathematical Journal》2011,62(9):1476-1486
We consider a relationship between two sets of extensions of a finite finitely additive measure μ defined on an algebra
\mathfrakB \mathfrak{B} of sets to a broader algebra
\mathfrakA \mathfrak{A} . These sets are the set ex S
μ
of all extreme extensions of the measure μ and the set H
μ
of all extensions defined as
l(A) = [^(m)]( h(A) ), A ? \mathfrakA \lambda (A) = \hat{\mu }\left( {h(A)} \right),\,\,\,A \in \mathfrak{A} , where [^(m)] \hat{\mu } is a quotient measure on the algebra
\mathfrakB