共查询到20条相似文献,搜索用时 31 毫秒
1.
T. I. Seidman 《Israel Journal of Mathematics》1969,7(3):249-253
Forλεσ(A) (A a bounded linear operator on a Hilbert space) withλ a boundary point of the numerical range, the ‘spectral theory’ forλ is ‘just as ifA were normal’. IfA isnormal-like (the smallest disk containingσ(A) has radiusr=inf
z
‖A − z‖), then also sup {‖Ax‖2 − |〈x.Ax〉|2:‖x‖=1}=r
2.
This research was partially supported by Air Force Contract AF-AFOSR-62-414. 相似文献
2.
In this paper, we study the perturbation bounds for the polar decomposition A= QH where Q is unitary and H is Hermitian. The optimal (asymptotic) bounds obtained in previous works for the unitary factor, the Hermitian factor and singular values of A are σ2r||△Q||2F ≤ ||△A||2F,1/2||△H||2F ≤ ||△A||2F and ||△∑||2F ≤ ||△A||2F, respectively, where ∑ = diag(σ1, σ2,..., σr, 0,..., 0) is the singular value matrix of A and σr denotes the smallest nonzero singular value. Here we present some new combined (asymptotic)perturbation bounds σ2r ||△Q||2F 1/2||△H||2F≤ ||△A||2F and σ2r||△Q||2F ||△∑ ||2F ≤||△A||2F which are optimal for each factor. Some corresponding absolute perturbation bounds are also given. 相似文献
3.
Pascale Vitse 《Rendiconti del Circolo Matematico di Palermo》2004,53(2):283-312
For Banach space operatorsT satisfying the Tadmor-Ritt condition ‖(zI−T)−1‖≤C|z−1|−1, |z|>1, we show how to use the Riesz turndown collar theorem to estimate sup
n≥0‖T
n‖. A similar estimate is shown for lim sup
n
‖T
n‖ in terms of the Ritt constantM=lim sup
z→1‖(1−z)(zI−T)−1‖. We also obtain an estimate of the functional calculus for these operators proving, in particular, that ‖f(T)‖≤C
q‖f‖
Mult
, where ‖·‖
Mult
stands for the multiplier norm of the Cauchy-Stieltjes integrals over a Lusin type cone domain depending onC and a parameterq, 0<q<1.
Notation.D denotes the open unit disc of the complex plane,D={z∈ℂ:|z|<1}, andT={z∈ℂ:|z|=1} is the unit circle.H
∞ is the Banach algebra of bounded analytic functions onD equipped with the supremum norm ‖.‖∞. 相似文献
4.
Rajendra Bhatia 《印度理论与应用数学杂志》2010,41(1):99-111
Lipschitz continuity of the matrix absolute value |A| = (A*A)1/2 is studied. Let A and B be invertible, and let M
1 = max(‖A‖, ‖B‖), M
2 = max(‖A
−1‖, ‖B
−1‖). Then it is shown that
$
\left\| { \left| A \right| - \left| B \right| } \right\| \leqslant \left( {1 + log M_1 M_2 } \right) \left\| {A - B} \right\|
$
\left\| { \left| A \right| - \left| B \right| } \right\| \leqslant \left( {1 + log M_1 M_2 } \right) \left\| {A - B} \right\|
相似文献
5.
L. Yu. Kolotilina 《Journal of Mathematical Sciences》2009,157(5):701-714
The paper presents a general approach to deriving inclusion sets for the singular values of a matrix A = (aij) ∈ ℂ
n×n. The key to the approach is the following result: If σ is a singular value of A, then a certain matrix C(σ, A) of order 2n,
whose diagonal entries are σ2 − | aii|2, i = 1, …, n, is singular. Based on this result, we use known diagonal-dominance type nonsingularity conditions to obtain
inclusion sets for the singular values of A. Scaled versions of the inclusion sets, allowing one, in particular, to obtain
Ky Fan type results for the singular values, are derived by passing to the conjugated matrix D−1C(σ, A)D, where D is a positive-definite diagonal matrix. Bibliography: 16 titles.
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 359, 2008, pp. 52–77. 相似文献
6.
In this paper, the matrix of equidiagonal-dominance is defined and several theorems about ‖A
−1‖∞ and its evaluation are established. Many interesting numerical examples are given.
This work is supported by the National Natural Science Foundation of China and the Science Foundation of Academy of Engineering
Physics of China 相似文献
7.
We develop new solvability methods for divergence form second order, real and complex, elliptic systems above Lipschitz graphs,
with L
2 boundary data. The coefficients A may depend on all variables, but are assumed to be close to coefficients A
0 that are independent of the coordinate transversal to the boundary, in the Carleson sense ‖A−A
0‖
C
defined by Dahlberg. We obtain a number of a priori estimates and boundary behaviour results under finiteness of ‖A−A
0‖
C
. Our methods yield full characterization of weak solutions, whose gradients have L
2 estimates of a non-tangential maximal function or of the square function, via an integral representation acting on the conormal
gradient, with a singular operator-valued kernel. Also, the non-tangential maximal function of a weak solution is controlled
in L
2 by the square function of its gradient. This estimate is new for systems in such generality, and even for real non-symmetric
equations in dimension 3 or higher. The existence of a proof a priori to well-posedness, is also a new fact. 相似文献
8.
L. Yu. Glebskii 《Mathematical Notes》1999,65(1):31-40
Theorems are proved establishing a relationship between the spectra of the linear operators of the formA+Ωg
iBigi
−1 andA+B
i, whereg
i∈G, andG is a group acting by linear isometric operators. It is assumed that the closed operatorsA andB
i possess the following property: ‖B
iA−1gBjA−1‖→0 asd(e,g)→∞. Hered is a left-invariant metric onG ande is the unit ofG. Moreover, the operatorA is invariant with respect to the action of the groupG. These theorems are applied to the proof of the existence of multicontour solutions of dynamical systems on lattices.
Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 37–47, January, 1999. 相似文献
9.
J. J. Grobler 《Israel Journal of Mathematics》1988,64(1):32-38
LetA be a unital Banach lattice algebra and leta εA
+ satisfy ‖a ‖≦1. Then either ‖a
n+1 −a
n ‖=2 for alln≧0 or else ‖a
n+1 −a
n ‖ → 0 asn → ∞. Cyclicity of the peripheral spectrum ofa is also established. 相似文献
10.
E is a Banach lattice that is weakly sequentially complete and has a weak unitu. TLf
n=ϕ means that the infimum of |f
n−ϕ| andu converges strongly to zero.T is a positive contraction operator onE andA
n=(1/n)(I+T+...+T
n−1). Without an additional assumption onE, the “truncated limit” TLA
nf need not exist forf inE. This limit exists for eachf ifE satisfies the following additional assumption (C): For everyf inE
+ and for every numberα>0, there is a numberβ=β(f, α) such that ifg is inE
+, ‖g‖≦1, 0≦f′≦f and ‖f′‖>α then ‖f′+g‖≧‖g‖+β.
Research of this author is partially supported by NSERC Grant A3974.
Research of this author is partially supported by NSF Grant 8301619. 相似文献
11.
The paper demonstrates that in computing a linear form (g, x) on the solution of a system of linear equations Ax = f, the
round-off error depends on the quantities ‖A−1f‖ and ‖AT−1g‖ rather than on the condition number of the coefficient matrix A. Estimates of the inherent and round-off errors in solving
the above problem by the orthogonalization method are provided. Numerical results confirming theoretical conclusions are presented.
Bibliography: 8 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 334, 2006, pp. 193–211. 相似文献
12.
Gerd Grubb 《Israel Journal of Mathematics》1971,10(1):32-95
The paper treats coerciveness inequalities (of the form Re(Au, u)≧c ‖u‖
s
2
−λ ‖u‖
0
2
,c>0,λ ∈ R) and semiboundedness inequalities (of the form Re (Au, u)≧−λ ‖u‖2) for the general boundary problems associated with an elliptic 2m-order differential operatorA in a compactn-dimensional manifold with boundary. In particular, we study the normal pseudo-differential boundary conditions, for which
we determine necessary and sufficient conditions for coerciveness withs=m, and for semiboundedness with ‖u‖ = ‖u‖m, in explicit form. 相似文献
13.
T. Sanders 《Journal d'Analyse Mathématique》2007,101(1):123-162
The paper has two main parts. To begin with, suppose that G is a compact abelian group. Chang’s Theorem can be viewed as a structural refinement of Bessel’s inequality for functions
ƒ ∈ L
2(G). We prove an analogous result for functions ƒ ∈ A(G), where A(G) is the space
endowed with the norm
, and generalize this to the approximate Fourier transform on Bohr sets.
As an application of the first part of the paper, we improve a recent result of Green and Konyagin. Suppose that p is a prime number and A ⊂ ℤ/pℤ has density bounded away from 0 and 1 by an absolute constant. Green and Konyagin have shown that ‖χ
A
‖
A(ℤ/pℤ) ≫ ɛ (log p)1/3−ɛ; we improve this to ‖χ
A
‖
A(ℤ/pℤ) ≫ ɛ (log p)1/2−ɛ. To put this in context, it is easy to see that if A is an arithmetic progression, then ‖χ
A
‖
A(ℤ/pℤ) ≪ log p. 相似文献
14.
Richard S. Varga 《Linear algebra and its applications》1976,14(3):211-217
In a recent paper by J.M. Varah, an upper bound for 6A-16∞ was determined, under the assumption that A is strictly diagonally dominant, and this bound was then used to obtain a lower bound for the smallest singular value for A. In this note, this upper bound for 6A-16∞ is sharpened, and extended to a wider class of matrices. This bound is then used to obtain an improved lower bound for the smallest singular value of a matrix. 相似文献
15.
Greg W. Anderson 《Israel Journal of Mathematics》2003,138(1):139-156
Forx ∈ ℝ
n
andp≥1 put ‖x‖
p
:=(n
−1Σ|x
i|
p
)1/p
. An orthogonal direct sum decomposition ℝ2k
=E⊕E
⊥ where dimE=k and
‖x‖2/‖x‖1≤C is called here a (k, C)-splitting. By a theorem of Kašin there existsC>0 such that (k, C)-splittings exist for allk, and by the volume ratio method of Szarek one can takeC=32eπ. All proofs of existence of (k, C)-splittings heretofore given are nonconstructive.
Here we investigate the representation of (k, C)-splittings by matrices with integral entries. For everyC>8e
1/2
π
−1/2 and positive integerk we specify a positive integerN(k, C) such that in the set ofk by 2k matrices with integral entries of absolute value not exceedingN(k, C) there exists a matrix with row span a summand in a (k, C)-splitting. We haveN(k, C)≤218k
fork large enough depending onC. We explain in detail how to test a matrix for the property of representing a (k, C)-splitting. Taken together our results yield an explicit (if impractical) construction of (k, C)-splittings. 相似文献
16.
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
17.
Rumi Shindo 《Central European Journal of Mathematics》2010,8(1):135-147
Let A and B be uniform algebras. Suppose that α ≠ 0 and A
1 ⊂ A. Let ρ, τ: A
1 → A and S, T: A
1 → B be mappings. Suppose that ρ(A
1), τ(A
1) and S(A
1), T(A
1) are closed under multiplications and contain expA and expB, respectively. If ‖S(f)T(g) − α‖∞ = ‖ρ(f)τ(g) − α‖∞ for all f, g ∈ A
1, S(e
1)−1 ∈ S(A
1) and S(e
1) ∈ T(A
1) for some e
1 ∈ A
1 with ρ(e
1) = 1, then there exists a real-algebra isomorphism $
\tilde S
$
\tilde S
: A → B such that $
\tilde S
$
\tilde S
(ρ(f)) = S(e
1)−1
S(f) for every f ∈ A
1. We also give some applications of this result. 相似文献
18.
In this paper, we obtain a version of subordination lemma for hyperbolic disk relative to hyperbolic geometry on the unit disk D. This subordination lemma yields the distortion theorem for Bloch mappings f ∈ H(B^n) satisfying ||f||0 = 1 and det f'(0) = α ∈ (0, 1], where||f||0 = sup{(1 - |z|^2 )n+1/2n det(f'(z))[1/n : z ∈ B^n}. Here we establish the distortion theorem from a unified perspective and generalize some known results. This distortion theorem enables us to obtain a lower bound for the radius of the largest univalent ball in the image of f centered at f(0). When a = 1, the lower bound reduces to that of Bloch constant found by Liu. When n = 1, our distortion theorem coincides with that of Bonk, Minda and Yanagihara. 相似文献
19.
Rainer Wittmann 《Israel Journal of Mathematics》1987,59(1):8-28
LetT be a positive linear contraction inL
p (1≦p<∞), then we show that lim ‖T
pf −T
n+1
f‖
p
≦(1 − ε)21/p
(f∈L
p
+
, ε>0 independent off) implies already limn
n→∞ ‖T
nf −T
n+1
n+1f ‖p
p=0. Several other related results as well as uniform variants of these are also given. Finally some similar results inLsu/t8 andC(X) are shown. 相似文献
20.
Several improvements are made to an algorithm of Higham and Smith for computing the matrix cosine. The original algorithm
scales the matrix by a power of 2 to bring the ∞-norm to 1 or less, evaluates the [8/8] Padé approximant, then uses the double-angle
formula cos (2A)=2cos 2A−I to recover the cosine of the original matrix. The first improvement is to phrase truncation error bounds in terms of ‖A2‖1/2 instead of the (no smaller and potentially much larger quantity) ‖A‖. The second is to choose the degree of the Padé approximant to minimize the computational cost subject to achieving a desired
truncation error. A third improvement is to use an absolute, rather than relative, error criterion in the choice of Padé approximant;
this allows the use of higher degree approximants without worsening an a priori error bound. Our theory and experiments show
that each of these modifications brings a reduction in computational cost. Moreover, because the modifications tend to reduce
the number of double-angle steps they usually result in a more accurate computed cosine in floating point arithmetic. We also
derive an algorithm for computing both cos (A) and sin (A), by adapting the ideas developed for the cosine and intertwining the cosine and sine double angle recurrences.
AMS subject classification 65F30
Numerical Analysis Report 461, Manchester Centre for Computational Mathematics, February 2005.
Gareth I. Hargreaves: This work was supported by an Engineering and Physical Sciences Research Council Ph.D. Studentship.
Nicholas J. Higham: This work was supported by Engineering and Physical Sciences Research Council grant GR/T08739 and by a
Royal Society–Wolfson Research Merit Award. 相似文献
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