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1.
We consider the Schr?dinger operator Hγ = ( − Δ)l + γ V(x)· acting in the space
where 2l ≥ d, V (x) ≥ 0, V (x) is continuous and is not identically zero, and
We study the asymptotic behavior as
of the non-bottom negative eigenvalues of Hγ, which are born at the moment γ = 0 from the lower bound λ = 0 of the spectrum σ(H0) of the unperturbed operator H0 = ( − Δ)l (virtual eigenvalues). To this end we use the Puiseux-Newton diagram for a power expansion of eigenvalues of some class of
polynomial matrix functions. For the groups of virtual eigenvalues, having the same rate of decay, we obtain asymptotic estimates
of Lieb-Thirring type. 相似文献
2.
Let T be an M-hyponormal operator acting on infinite dimensional separable Hilbert space and let
be the Riesz idempotent for λ0, where D is a closed disk of center λ0 which contains no other points of σ (T). In this note we show that E is self-adjoint and
As an application, if T is an algebraically M-hyponormal operator then we prove : (i) Weyl’s theorem holds for f(T) for every
(ii) a-Browder’s theorem holds for f(S) for every
and f ∈ H(σ(S)); (iii) the the spectral mapping theorem holds for the Weyl spectrum of T and for the essential approximate point spectrum of T. 相似文献
3.
A detailed study is made of matrix-valued, ordinary linear differential operators T in for 1 < p < ∞, which arise as the perturbation of a constant coefficient differential operator of order n ≥ 1 by a lower order differential operator which has a factorisation S = AB for suitable operators A and B. Via techniques from L
p
-harmonic analysis, perturbation theory and local spectral theory, it is shown that T satisfies certain local resolvent estimates, which imply the existence of local functional calculi and decomposability properties
of T.
相似文献
4.
Luciana Angiuli Michele MirandaJr Diego Pallara Fabio Paronetto 《Annali di Matematica Pura ed Applicata》2009,188(2):297-331
Given a uniformly elliptic second order operator on a possibly unbounded domain , let (T(t))
t≥0 be the semigroup generated by in L
1(Ω), under homogeneous co-normal boundary conditions on ∂Ω. We show that the limit as t → 0 of the L
1-norm of the spatial gradient D
x
T(t)u
0 tends to the total variation of the initial datum u
0, and in particular is finite if and only if u
0 belongs to BV(Ω). This result is true also for weighted BV spaces. A further characterization of BV functions in terms of the short-time behaviour of (T(t))
t≥0 is also given.
相似文献
5.
We consider the Schr?dinger operator Hγ = ( − Δ)l + γ V(x)· acting in the space
$$L_2 (\mathbb{R}^d ),$$ where 2l ≥ d, V (x) ≥ 0, V (x) is continuous and is not identically zero, and
$$\lim _{|{\mathbf{x}}| \to \infty } V({\mathbf{x}}) = 0.$$ We obtain an asymptotic expansion as
$$\gamma \uparrow 0$$of the bottom negative eigenvalue of Hγ, which is born at the moment γ = 0 from the lower bound λ = 0 of the spectrum σ(H0) of the unperturbed operator H0 = ( − Δ)l (a virtual eigenvalue). To this end we develop a supplement to the Birman-Schwinger theory on the process of the birth of
eigenvalues in the gap of the spectrum of the unperturbed operator H0. Furthermore, we extract a finite-rank portion Φ(λ) from the Birman- Schwinger operator
$$X_V (\lambda ) = V^{\frac{1} {2}} R_\lambda (H_0 )V^{\frac{1}{2}} ,$$ which yields the leading terms for the desired asymptotic
expansion. 相似文献
6.
Stephan Ramon Garcia 《Integral Equations and Operator Theory》2008,60(3):357-367
If denotes the polar decomposition of a bounded linear operator T, then the Aluthge transform of T is defined to be the operator . In this note we study the relationship between the Aluthge transform and the class of complex symmetric operators (T iscomplex symmetric if there exists a conjugate-linear, isometric involution so that T = CT*C). In this note we prove that: (1) the Aluthge transform of a complex symmetric operator is complex symmetric, (2) if T is complex symmetric, then and are unitarily equivalent, (3) if T is complex symmetric, then if and only if T is normal, (4) if and only if T
2 = 0, and (5) every operator which satisfies T
2 = 0 is necessarily complex symmetric.
This work partially supported by National Science Foundation Grant DMS 0638789. 相似文献
7.
Horst R. Thieme 《Journal of Evolution Equations》2008,8(2):283-305
If T = {T (t); t ≥ 0} is a strongly continuous family of bounded linear operators between two Banach spaces X and Y and f ∈ L
1(0, b, X), the convolution of T with f is defined by . It is shown that T * f is continuously differentiable for all f ∈ C(0, b, X) if and only if T is of bounded semi-variation on [0, b]. Further T * f is continuously differentiable for all f ∈ L
p
(0, b, X) (1 ≤ p < ∞) if and only if T is of bounded semi-p-variation on [0, b] and T(0) = 0. If T is an integrated semigroup with generator A, these respective conditions are necessary and sufficient for the Cauchy problem u′ = Au + f, u(0) = 0, to have integral (or mild) solutions for all f in the respective function vector spaces. A converse is proved to a well-known result by Da Prato and Sinestrari: the generator
A of an integrated semigroup is a Hille-Yosida operator if, for some b > 0, the Cauchy problem has integral solutions for all f ∈ L
1(0, b, X). Integrated semigroups of bounded semi-p-variation are preserved under bounded additive perturbations of their generators and under commutative sums of generators
if one of them generates a C
0-semigroup.
Günter Lumer in memoriam 相似文献
8.
A. M. Gomilko 《Ukrainian Mathematical Journal》2004,56(8):1212-1226
We consider the problem of estimates for the powers of the Cayley transform V = (A + I)(A - I)–1 of the generator of a uniformly bounded C
0-semigroup of operators e
tA
, t 0, that acts in a Hilbert space H. In particular, we establish the estimate
. We show that the estimate
is true in the following cases: (a) the semigroups e
tA
and
are uniformly bounded; (b) the semigroup e
tA
uniformly bounded for t is analytic (in particular, if the generator of the semigroup is a bounded operator).Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 8, pp. 1018–1029, August, 2004. 相似文献
9.
In this paper, we prove that if a sequence of homeomorphisms , with bounded planar domains, of Sobolev space has uniformly equibounded distortions in EXP(Ω) and weakly converges to f in then the matrices A(x, f
j
) of the corresponding Laplace-Beltrami operators Γ-converge in the Orlicz–Sobolev space , where Q(t) = t
2log(e + t), to the matrix A(x, f) of the Laplace-Beltrami operator associated to f.
相似文献
10.
Adam Bobrowski 《Journal of Evolution Equations》2007,7(3):555-565
Let
be a locally compact Hausdorff space. Let A and B be two generators of Feller semigroups in
with related Feller processes {X
A
(t), t ≥ 0} and {X
B
(t), t ≥ 0} and let α and β be two non-negative continuous functions on
with α + β = 1. Assume that the closure C of C
0 = αA + βB with
generates a Feller semigroup {T
C
(t), t ≥ 0} in
. It is natural to think of a related Feller process {X
C
(t), t ≥ 0} as that evolving according to the following heuristic rules. Conditional on being at a point
, with probability α(p) the process behaves like {X
A
(t), t ≥ 0} and with probability β(p) it behaves like {X
B
(t), t ≥ 0}. We provide an approximation of {T
C
(t), t ≥ 0} via a sequence of semigroups acting in
that supports this interpretation. This work is motivated by the recent model of stochastic gene expression due to Lipniacki
et al. [17]. 相似文献
11.
Yisheng Song 《Positivity》2009,13(4):643-655
In this paper, for a Lipschitz pseudocontractive mapping T, we study the strong convergence of iterative schemes generated by
, where f is a Lipschitz strong pseudocontractive mapping and {βn}, {αn} satisfy (i); (ii) ; (iii).
相似文献
12.
Jong-Do Park 《Integral Equations and Operator Theory》2006,54(4):571-584
We investigate necessary and sufficient conditions for boundedness of the operator
on the Bergman space of the unit ball
for n ≥ 1, where Tf is the Toeplitz operator. Those conditions are related to boundedness of the Berezin transform of symbols f and g. We construct the inner product formula which plays a crucial role in proving the sufficiency of the conditions. 相似文献
13.
B. P. Duggal 《Integral Equations and Operator Theory》2009,63(1):17-28
A Banach space operator T ∈ B(χ) is polaroid if points λ ∈ iso σ(T) are poles of the resolvent of T. Let denote, respectively, the approximate point, the Weyl, the Weyl essential approximate, the upper semi–Fredholm and lower
semi–Fredholm spectrum of T. For A, B and C ∈ B(χ), let M
C
denote the operator matrix . If A is polaroid on , M
0 satisfies Weyl’s theorem, and A and B satisfy either of the hypotheses (i) A has SVEP at points and B has SVEP at points , or, (ii) both A and A* have SVEP at points , or, (iii) A* has SVEP at points and B
* has SVEP at points , then . Here the hypothesis that λ ∈ π0(M
C
) are poles of the resolvent of A can not be replaced by the hypothesis are poles of the resolvent of A.
For an operator , let . We prove that if A* and B* have SVEP, A is polaroid on π
a
0(M
C) and B is polaroid on π
a
0(B), then .
相似文献
14.
Gaspar Mora 《Mediterranean Journal of Mathematics》2005,2(3):315-325
Given a real function f of class
defined on the unit cube In=[0,1]n , n ≥ 2, our purpose consists in finding an algorithm to approximate to
by a dimensional reduction. The method deals with α-dense curves γα in the domain In with arbitrary small density α and a minimization-preserving operator T (briefly M.P.O.) applied to the univariable function
By reiterating the action of this M.P.O. we obtain an algorithm to determine a global minimizer t0* of fα. The value fα(t0*), taken as an approximation to f*, only depends on the density α of the curve chosen to densify the domain of the objective function. 相似文献
15.
Yuri I. Karlovich Enrique Ramírez de Arellano 《Integral Equations and Operator Theory》2001,39(4):441-474
LetB be the Banach algebra of all bounded linear operators on the weighted Lebesgue spaceL
p (T, ) with an arbitrary Muckenhoupt weight on the unit circleT, and
the Banach subalgebra ofB generated by the operators of multiplication by piecewise continuous coefficients and the operatorse
h,S
T
e
h,
–1
I (hR, T) whereS
T is the Cauchy singular integral operator ande
h,(t)=exp(h(t+)/(t–)),tT. The paper is devoted to a symbol calculus, Fredholm criteria and an index formula for the operators in the algebra
and its matrix analogue
. These shift-invariant algebras arise naturally in studying the algebras of singular integral operators with coefficients admitting semi-almost periodic discontinuities and shifts being diffeomorphisms ofT onto itself with second Taylor derivatives.Partially supported by CONACYT grant, Cátedra Patrimonial, No. 990017-EX and by CONACYT project 32726-E, México. 相似文献
16.
Robert J. Taggart 《Mathematische Zeitschrift》2009,261(4):933-949
Suppose that {T
t
: t ≥ 0} is a symmetric diffusion semigroup on L
2(X) and denote by its tensor product extension to the Bochner space , where belongs to a certain broad class of UMD spaces. We prove a vector-valued version of the Hopf–Dunford–Schwartz ergodic theorem
and show that this extends to a maximal theorem for analytic continuations of on . As an application, we show that such continuations exhibit pointwise convergence. 相似文献
17.
Maribel Loaiza Marcos López-García Salvador Pérez-Esteva 《Integral Equations and Operator Theory》2005,53(2):287-296
In this paper we decompose
into diadic annuli
and consider the class Sp,q of Toeplitz operators Tφ for which the sequence of Schatten norms
belongs to ℓq, where φn = φχ An. We study the boundedness and compactness of the operators in Sp,q and we describe the operators Tφ , φ ≥ 0 in these spaces in terms of weighted Herz norms of the averaging operator of the symbols φ. 相似文献
18.
H. S. Mustafayev 《Integral Equations and Operator Theory》2007,57(2):235-246
Let G be a locally compact abelian group and let
be a representation of G by means of isometries on a Banach space. We define WT as the closure with respect to the weak operator topology of the set
where
is the Fourier transform of f ∈L1(G) with respect to the group T. Then WT is a commutative Banach algebra. In this paper we study semisimlicity problem for such algebras. The main result is that
if the Arveson spectrum sp(T) of T is scattered (i.e. it does not contain a nonempty perfect subset) then the algebra WT is semisimple.
Some related problems are also discussed. 相似文献
19.
We show that if the pseudodifferential operator −q(x,D) generates a Feller semigroup (Tt)t≥0 then the Feller semigroups (Tt(v))t≥0 generated by the pseudodifferential operators with symbol will converge strongly to (Tt)t≥0 as ν →∞. 相似文献
20.
Fausto Gozzi 《Journal of Evolution Equations》2006,6(4):711-743
We consider a semilinear stochastic differential equation in a Hilbert space H with a Lipschitz continuous (possibly unbounded) nonlinearity F. We prove that the associated transition semigroup {Pt, t ≥ 0}, acting on the space of bounded measurable functions from H to
, transforms bounded nondifferentiable functions into Fréchet differentiable ones. Moreover we consider the associated Kolmogorov
equation and we prove that it possesses a unique “strong” solution (where “strong” means limit of classical solutions) given
by the semigroup {Pt, t ≥ 0} applied to the initial condition. This result is a starting point to prove existence and uniqueness of strong solutions
to Hamilton - Jacobi - Bellman equations arising in control theory.
Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday 相似文献