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1.
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
2.
3.
A complex number λ is an extended eigenvalue of an operator A if there is a nonzero operator X such that AX = λ XA. We characterize the set of extended eigenvalues, which we call extended point spectrum, for operators acting on finite dimensional
spaces, finite rank operators, Jordan blocks, and C0 contractions. We also describe the relationship between the extended eigenvalues of an operator A and its powers. As an application, we show that the commutant of an operator A coincides with that of An, n ≥ 2, n ∈ N if the extended point spectrum of A does not contain any n–th root of unity other than 1. The converse is also true if either A or A* has trivial kernel. 相似文献
4.
Patrik Wahlberg 《Integral Equations and Operator Theory》2007,59(1):99-128
We study the short-time Fourier transformation, modulation spaces, Gabor representations and time-frequency localization operators,
for functions and tempered distributions that have as range space a Banach or a Hilbert space. In the Banach space case the
theory of modulation spaces contains some modifications of the scalar-valued theory, depending on the Banach space. In the
Hilbert space case the modulation spaces have properties similar to the scalar-valued case and the Gabor frame theory essentially
works. For localization operators in this context symbols are operator-valued. We generalize two results from the scalar-valued
theory on continuity on certain modulation spaces when the symbol belongs to an Lp,q space and M∞, respectively. The first result is true for any Banach space as range space, and the second result is true for any Hilbert
space as range space. 相似文献
5.
Anders Olofsson 《Integral Equations and Operator Theory》2007,58(4):503-549
We study an operator-valued Berezin transform corresponding to certain standard weighted Bergman spaces of square integrable
analytic functions in the unit disc. The study of this operator-valued Berezin transform relates in a natural way to the study
of the class of n-hypercontractions on Hilbert space introduced by Agler. To an n-hypercontraction
we associate a positive
-valued operator measure dω
n, T
supported on the closed unit disc
in a way that generalizes the above notion of operator-valued Berezin transform. This construction of positive operator measures
dω
n, T
gives a natural functional calculus for the class of n-hypercontractions. We revisit also the operator model theory for the class of n-hypercontractions. The new results here concern certain canonical features of the theory. The operator model theory for the
class of n-hypercontractions gives information about the structure of the positive operator measures dω
n, T
. 相似文献
6.
Representations are given for the multiplicity of an analytic operator-valued function A at an isolated point z0 of the spectrum in the form of kernels and ranges of Hankel and Toeplitz matrices whose entries are derived from the Taylor coefficients of A and the Laurent coefficients of A−1 about z0. In two special cases the results can be expressed in terms of finite matrices: when A is a polynomial and when A−1 has a pole at z0. The latter case leads to the theory of Jordan chains. 相似文献
7.
8.
Nathan S. Feldman 《Integral Equations and Operator Theory》2007,58(2):153-173
A pair of commuting operators, (A,B), on a Hilbert space
is said to be hypercyclic if there exists a vector
such that {A
n
B
k
x : n, k ≥ 0} is dense in
. If f, g ∈H
∞(G) where G is an open set with finitely many components in the complex plane, then we show that the pair (M
*
f
, M
*
g
) of adjoints of multiplcation operators on a Hilbert space of analytic functions on G is hypercyclic if and only if the semigroup they generate contains a hypercyclic operator. However, if G has infinitely many components, then we show that there exists f, g ∈H
∞(G) such that the pair (M
*
f
, M
*
g
) is hypercyclic but the semigroup they generate does not contain a hypercyclic operator. We also consider hypercyclic n-tuples. 相似文献
9.
We use operator-valued Fourier multiplier theorems to study second order differential equations in Banach spaces. We establish
maximal regularity results in Lp and Cs for strong solutions of a complete second order equation.
In the second part, we study mild solutions for the second order problem. Two types of mild solutions are considered. When
the operator A involved is the generator of a strongly continuous cosine function, we give characterizations in terms of Fourier multipliers
and spectral properties of the cosine function. The results obtained are applied to elliptic partial differential operators.
The first author is supported in part by Convenio de Cooperación Internacional (CONICYT) Grant # 7010675 and the second author
is partially financed by FONDECYT Grant # 1010675 相似文献
10.
Jean-Pierre Borel 《Israel Journal of Mathematics》1988,64(2):229-250
For a given subsetA of the set of real numbers, we search a sequence Λ=(λ
n) of real numbers such that bothA is the normal setB(Λ) associated to Λ, and Λ takes its values in a bounded interval, with a minimal lengthM. A lower bound ofM is obtained, which gives some necessary conditions of existency of such a bounded sequence Λ. More details are given whenA is a subset of the set of integers. In this case, the problem is to find a polynomialQ of lowest degree such that the productP.Q has non-negative coefficients, for some special given polynomialP.
相似文献
11.
Oscar F. Bandtlow 《Integral Equations and Operator Theory》2008,61(1):21-43
For a, α > 0 let E(a, α) be the set of all compact operators A on a separable Hilbert space such that s
n
(A) = O(exp(-anα)), where s
n
(A) denotes the n-th singular number of A. We provide upper bounds for the norm of the resolvent (zI − A)−1 of A in terms of a quantity describing the departure from normality of A and the distance of z to the spectrum of A. As a consequence we obtain upper bounds for the Hausdorff distance of the spectra of two operators in E(a, α).
相似文献
12.
H. H. Schaefer 《Israel Journal of Mathematics》1984,48(2-3):196-204
Let (X, Σ, μ) be a σ-finite measure space,T a compact irreducible (positive, linear) operator onL
p (μ) (1≦p<+∞). It is shown that the spectral radiusr ofT is characterized by the minimax property {fx196-1} where ∑0 denotes the ring of sets of finite measure and whereQ denotes the set of all, almost everywhere positive functions inL
p. Moreover, ifr>0 then equality on either side is assumed ifff is the (essentially unique) positive eigenfunction ofT. Various refinements are given in terms of corresponding relations for irreducible finite rank operators approximatingT.
Dedicated to H. G. Tillmann on his 60th birthday 相似文献
13.
Bhagwati Prashad Duggal Slavisa V. Djordjević 《Mediterranean Journal of Mathematics》2005,2(4):395-406
It is known that if
and
are Banach space operators with the single-valued extension property, SVEP, then the matrix operator
has SVEP for every operator
and hence obeys Browder’s theorem. This paper considers conditions on operators A, B, and M0 ensuring Weyls theorem for operators MC. 相似文献
14.
Ioan Rasa 《Mediterranean Journal of Mathematics》2005,2(2):153-169
Consider the Voronovskaja operator A of a sequence of positive linear operators
and let u(t, x) be the solution of the Cauchy problem for A. In the spirit of Altomare’s theory this solution can be studied by using the semigroup (T(t))t ≥ 0 generated by A and represented in terms of the operators Ln.One associates to A a stochastic equation; its solution can be also used in order to represent u(t, x).The relations between all these objects are described in the case of the operator A associated with some Meyer-König and Zeller type operators. 相似文献
15.
Wolfgang Hackenbroch 《manuscripta mathematica》1990,66(1):351-357
Given two σ-algebrasU ⊂A, invariant under a fixed semigroupG of transformations, the following subsetC of the lattice coneM (U)
G
ofG-invariant finite measures onU is shown to be (the positive part of) a band inM (U)
G
: AG-invariant measure μ belongs toC iff the setexM
Bμ)
G
of extremalG-invariant extensions of μ toB is non-empty and eachG-invariant extensionv of μ admits a barycentric decompositionv=→v′ρ(dv′) with some representing probability ρ onexM
U μ)
G
.—Any band of extensible measures allows to study the corresponding extension problem locally. 相似文献
16.
In this paper, we first give some invariant subspace results for collectively compact sets of operators in connection with
the joint spectral radius of these sets. We then prove that any collectively compact set M in algΓ satisfies Berger-Wang formula, where Γ is a complete chain of subspaces of X.
相似文献
17.
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
18.
Bernhard H. Haak Peer Christian Kunstmann 《Integral Equations and Operator Theory》2006,55(4):497-533
We study linear systems, described by operators A, B, C for which the state space X is a Banach space.We suppose that − A generates a bounded analytic semigroup and give conditions for admissibility of B and C corresponding to those in G. Weiss’ conjecture. The crucial assumptions on A are boundedness of an H∞-calculus or suitable square function estimates, allowing to use techniques recently developed by N. Kalton and L. Weis. For observation spaces Y or control spaces U that are not Hilbert spaces we are led to a notion of admissibility extending previous considerations by C. Le Merdy. We also obtain a characterisation of wellposedness for the full system. We give several examples for admissible operators
including point observation and point control. At the end we study a heat equation in X = Lp(Ω), 1 < p < ∞, with boundary observation and control and prove its wellposedness for several function spaces Y and U on the boundary ∂Ω. 相似文献
19.
Functions whose values are bounded linear Hilbert space operators (each operator may be defined on its own subspace of the
ambient Hilbert space), the domain of definition is contained in the open unit disc, and having the following property κ, are studied. (κ): All Pick operators associated with the function have the dimensions of their spectral subspace corresponding to the negative
part of the spectrum bounded above by a fixed nonnegative integer κ, and the bound κ is attained. No a priori hypotheses concerning regularity of the functions are assumed. A particular class of functions, called standard
functions, is introduced, and the corresponding nonnegative integer κ is identified for standard functions. It is proved that every function with property (κ) can be extended to a standard function with property (κ), for the same κ. This result is interpreted as a result on interpolation. As an application, maximal (with respect to the extension relation)
functions with the property κ, for a fixed κ, are studied in terms of standard functions.
Received: August 5, 2007., Accepted: October 24, 2007. 相似文献
20.
A bounded linear operator T ∈ L(X) on aBanach space X is said to satisfy “Browder’s theorem” if the Browder spectrum coincides with the Weyl spectrum. T ∈ L(X) is said to satisfy “a-Browder’s theorem” if the upper semi-Browder spectrum coincides with the approximate point Weyl spectrum. In this note we give several characterizations of operators satisfying these theorems. Most of these characterizations are obtained by using a localized version of the single-valued extension property of T. In the last part we shall give some characterizations of operators for which “Weyl’s theorem” holds. 相似文献