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1.
Sergio Albeverio Alexander K. Motovilov Andrei A. Shkalikov 《Integral Equations and Operator Theory》2009,64(4):455-486
Let A be a self-adjoint operator on a Hilbert space . Assume that the spectrum of A consists of two disjoint components σ0 and σ1. Let V be a bounded operator on , off-diagonal and J-self-adjoint with respect to the orthogonal decomposition where and are the spectral subspaces of A associated with the spectral sets σ0 and σ1, respectively. We find (optimal) conditions on V guaranteeing that the perturbed operator L = A + V is similar to a self-adjoint operator. Moreover, we prove a number of (sharp) norm bounds on the variation of the spectral
subspaces of A under the perturbation V. Some of the results obtained are reformulated in terms of the Krein space theory. As an example, the quantum harmonic oscillator
under a -symmetric perturbation is discussed.
This work was supported by the Deutsche Forschungsgemeinschaft (DFG), the Heisenberg-Landau Program, and the Russian Foundation
for Basic Research. 相似文献
2.
We study the self-adjoint and dissipative realization A of a second order elliptic differential operator
with unbounded regular coefficients in
, where μ(dx) = ρ (x)dx is the associated invariant measure. We prove a maximal regularity result under suitable assumptions, that generalize the
well known conditions in the case of constant diffusion part.
Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday 相似文献
3.
Heinz Langer Alexander Markus Vladimir Matsaev 《Integral Equations and Operator Theory》2009,63(4):533-545
In this note we continue the study of spectral properties of a self-adjoint analytic operator function A(z) that was started in [5]. It is shown that if A(z) satisfies the Virozub–Matsaev condition on some interval Δ0 and is boundedly invertible in the endpoints of Δ0, then the ‘embedding’ of the original Hilbert space into the Hilbert space , where the linearization of A(z) acts, is in fact an isomorphism between a subspace of and . As a consequence, properties of the local spectral function of A(z) on Δ0 and a so-called inner linearization of the operator function A(z) in the subspace are established.
相似文献
4.
The aim of this paper is to give the basic principles of hyperbolic function theory on the Clifford algebra . The structure of the theory is quite similar to the case of Clifford algebras with negative generators, but the proofs are
not obvious. The (real) Clifford algebra is generated by unit vectors with positive squares e2i = + 1. The hyperbolic Dirac operator is of the form where Q0f is represented by the composition . If is a solution of Hkf = 0, then f is called k-hypergenic in Ω, where is an open set. We introduce some basic results of hyperbolic function theory and give some representation theorems on .
Received: October, 2007. Accepted: February, 2008. 相似文献
5.
If E is a separable symmetric sequence space with trivial Boyd indices and
is the corresponding ideal of compact operators, then there exists a C1-function fE, a self-adjoint element
and a densely defined closed symmetric derivation δ on
such that
, but
相似文献
6.
Antonio G. García Miguel A. Hernández-Medina 《Mediterranean Journal of Mathematics》2005,2(3):345-356
Let
be a symmetric operator with compact resolvent defined in a Hilbert space
For any fixed
we consider an entire
function Ka which involves the resolvent of
Associated with Ka we obtain, by duality in
a Hilbert space
of entire functions which becomes a De Branges space of entire functions. This property provides a characterization of
regardless of the anti-linear mapping which has
as its range space. There exists also a sampling formula allowing to recover any function in
from its samples at the sequence of eigenvalues of
This work has been supported by the grant BFM2003–01034 from the D.G.I. of the Spanish Ministerio de Ciencia y Tecnología. 相似文献
7.
Here we study complete rotation hypersurfaces with constant k-th mean curvature Hk in
even and 2 < k < n. We prove the existence of a constant
such that there are no such hypersurfaces for
. We have only one compact hypersurface of this kind with
. For each
there is a corresponding family of complete immersed rotation hypersurfaces, each family containing two isoparametric hypersurfaces.
For Hk ≥ 0, there is also such a family, now containing only one isoparametric hypersurface. Finally, we prove the existence of
compact hypersurfaces with arbitrarily large Hk , neither isometric to a sphere nor to a product of spheres.
*Bull. Braz. Math. Soc. 30 (2), 1999, 139–161.
**Partially supported by FUNCAP, Brazil.
***Partially supported by CNPq, Brazil and DGAPA-UNAM, México. 相似文献
8.
9.
Hidetoshi Maeda 《Archiv der Mathematik》2007,88(5):419-424
Let
be an ample vector bundle of rank n – 1 on a smooth complex projective variety X of dimension n≥ 3 such that X is a
-bundle over
and that
for any fiber F of the bundle projection
. The pairs
with
= 2 are classified, where
is the curve genus of
. This allows us to improve some previous results.
Received: 13 June 2006 相似文献
10.
It is known [6] that for every function f in the generalized Schur class
and every nonempty open subset Ω of the unit disk
, there exist points z1,...,zn ∈Ω such that the n × nPick matrix
has κ negative eigenvalues. In this paper we discuss existence of an integer n0 such that any Pick matrix based on z1,...,zn ∈Ω with n ≥ n0 has κ negative eigenvalues. Definitely, the answer depends on Ω. We prove that if
, then such a number n0 does not exist unless f is a ratio of two finite Blaschke products; in the latter case the minimal value of n0 can be found. We show also that if the closure of Ω is contained in
then such a number n0 exists for every function f in
. 相似文献
11.
The main results of this paper are a generalization of the results of S. Fajtlowicz and J. Mycielski on convex linear forms.
We show that if Vn is the variety generated by all possible algebras
, where R denotes the real numbers and
, for some
, then any basis for the set of all identities satisfied by Vn is infinite. But on the other hand, the identities satisfied by Vn are a consequence of gL and μn, where μn is the n-ary medial law and the inference rule gL is an implication patterned after the classical rigidity lemma of algebraic geometry. We also prove that the identities satisfied
by
are a consequence of gL and μn iff {p1, ... , pn} is algebraically independent. We then prove analagous results for algebras
of arbitrary type τ and in the final section of this paper, we show that analagous results hold for Abelian group hyperidentities.
This paper is dedicated to Walter Taylor.
Received July 16, 2005; accepted in final form January 12, 2006.
The research of both authors was supported by an operating grant ODP0008215 from NSERC. 相似文献
12.
Jackson's Theorem on Bounded Symmetric Domains 总被引:1,自引:0,他引:1
Ming Zhi WANG Guang Bin REN 《数学学报(英文版)》2007,23(8):1391-1404
Polynomial approximation is studied on bounded symmetric domain Ω in C^n for holomorphic function spaces X such as Bloch-type spaces, Bergman-type spaces, Hardy spaces, Ω algebra and Lipschitz space. We extend the classical Jackson's theorem to several complex variables:Eκ(f,X)≤ω(1/k,f,X), where Eκ(f,X) is the deviation of the best approximation of f ∈X by polynomials of degree at most k with respect to the X-metric and ω(1/k,f,X) is the corresponding modulus of continuity. 相似文献
13.
Yah SUN Gen Xiang CHAI 《数学学报(英文版)》2007,23(4):685-696
In this article the following random intercept mixed effects model will be considered: yij = vi =v^τijβ+ εij,i=1,…,m;j=1,2,…,ni, where {vi} are i.i.d, random effects with mean α 2. 2 and finite variance σ^2 v, {εij} are i.i.d, random errors with finite variance ε^2 ε. Here we will estimate α,σ^2 v,σ^2 ε,β and study their large sample properties, such as strong consistency, strong convergence rates and asymptotic normality. 相似文献
14.
For a mixed hypergraph
, where
and
are set systems over the vertex set X, a coloring is a partition of X into ‘color classes’ such that every
meets some class in more than one vertex, and every
has a nonempty intersection with at least two classes. The feasible set of
, denoted
, is the set of integers k such that
admits a coloring with precisely k nonempty color classes. It was proved by Jiang et al. [Graphs and Combinatorics 18 (2002), 309–318] that a set S of natural numbers is the feasible set of some mixed hypergraph if and only if either
or S is an ‘interval’
for some integer k ≥ 1.
In this note we consider r-uniform mixed hypergraphs, i.e. those with |C| = |D| = r for all
and all
, r ≥ 3. We prove that S is the feasible set of some r-uniform mixed hypergraph with at least one edge if and only if either
for some natural number k ≥ r − 1, or S is of the form
where S′′ is any (possibly empty) subset of
and S′ is either the empty set or {r − 1} or an ‘interval’ {k, k + 1, ..., r − 1} for some k, 2 ≤ k ≤ r − 2. We also prove that all these feasible sets
can be obtained under the restriction
, i.e. within the class of ‘bi-hypergraphs’.
Research supported in part by the Hungarian Scientific Research Fund, OTKA grant T-049613. 相似文献
15.
Let
be the 2k-uniform hypergraph obtained by letting P1, . . .,Pr be pairwise disjoint sets of size k and taking as edges all sets Pi∪Pj with i ≠ j. This can be thought of as the ‘k-expansion’ of the complete graph Kr: each vertex has been replaced with a set of size k. An example of a hypergraph with vertex set V that does not contain
can be obtained by partitioning V = V1 ∪V2 and taking as edges all sets of size 2k that intersect each of V1 and V2 in an odd number of elements. Let
denote a hypergraph on n vertices obtained by this construction that has as many edges as possible. For n sufficiently large we prove a conjecture of Frankl, which states that any hypergraph on n vertices that contains no
has at most as many edges as
.
Sidorenko has given an upper bound of
for the Tur′an density of
for any r, and a construction establishing a matching lower bound when r is of the form 2p+1. In this paper we also show that when r=2p+1, any
-free hypergraph of density
looks approximately like Sidorenko’s construction. On the other hand, when r is not of this form, we show that corresponding constructions do not exist and improve the upper bound on the Turán density
of
to
, where c(r) is a constant depending only on r.
The backbone of our arguments is a strategy of first proving approximate structure theorems, and then showing that any imperfections
in the structure must lead to a suboptimal configuration. The tools for its realisation draw on extremal graph theory, linear
algebra, the Kruskal–Katona theorem and properties of Krawtchouck polynomials.
* Research supported in part by NSF grants DMS-0355497, DMS-0106589, and by an Alfred P. Sloan fellowship. 相似文献
16.
We consider a question raised by Suhov and Voice from quantum information theory and quantum computing. An element of a partition
of {1, ..., n} is said to be block-stable for
if it is not moved to another block under the action of π. The problem concerns the determination of the generating series
for elements of
with respect to the number of block-stable elements of a canonical partition of a finite n-set, with block sizes k1, ..., kr, in terms of the moment (power) sums pq(k1, ..., kr). We also consider the limit
subject to the condition that
exists for q = 1, 2,....
Received January 31, 2006 相似文献
17.
S. P. Eveson 《Integral Equations and Operator Theory》2005,53(3):331-341
Given k ∈ L1 (0,1) satisfying certain smoothness and growth conditions at 0, we consider the Volterra convolution operator Vk defined on Lp (0,1) by
and its iterates
We construct some much simpler sequences which, as n → ∞, are asymptotically equal in the operator norm to Vkn. This leads to a simple asymptotic formula for ||Vkn|| and to a simple ‘asymptotically extremal sequence’; that is, a sequence (un) in Lp (0, 1) with ||un||p=1 and
as n → ∞. As an application, we derive a limit theorem for large deviations, which appears to be beyond the established theory. 相似文献
18.
Xianling Fan Shao-Gao Deng 《NoDEA : Nonlinear Differential Equations and Applications》2009,16(2):255-271
We study the existence and multiplicity of positive solutions for the inhomogeneous Neumann boundary value problems involving
the p(x)-Laplacian of the form
where Ω is a bounded smooth domain in , and p(x) > 1 for with and φ ≢ 0 on ∂Ω. Using the sub-supersolution method and the variational method, under appropriate assumptions on f, we prove that, there exists λ* > 0 such that the problem has at least two positive solutions if λ = λ*, has at least one positive solution if λ = λ*, and has no positive solution if λ = λ*. To prove the result we establish a special strong comparison principle for the Neumann problems.
The research was supported by the National Natural Science Foundation of China 10371052,10671084). 相似文献
19.
We propose a unifying framework for studying extremal problems related to graph minors. This framework relates the existence
of a large minor in a given graph to its expansion properties. We then apply the developed framework to several extremal problems
and prove in particular that: (a) Every -free graph G with average degree r ( are constants) contains a minor with average degree , for some constant ; (b) Every C2k-free graph G with average degree r (k ≥ 2 is a constant) contains a minor with average degree , for some constant c = c(k) > 0. We also derive explicit lower bounds on the minor density in random, pseudo-random and expanding graphs.
Received: March 2008, Accepted: May 2008 相似文献
20.
Summary.
Let
We say that
preserves the distance d 0 if
for each
implies
Let A
n
denote the set of all positive numbers
d such that any map
that preserves unit distance preserves also distance
d.
Let D
n
denote the set of all positive numbers
d with the property: if
and
then there exists a finite set
S
xy
with
such that any map
that preserves unit distance preserves also the distance between
x and y.
Obviously,
We prove:
(1)
(2)
for n 2
D
n
is a
dense subset of
(2) implies that each mapping
f
from
to
(n 2)
preserving unit distance preserves all distances,
if f is continuous with respect to the product topologies
on
and
相似文献