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1.
Let X be a smooth manifold with boundary of dimension n > 1. The operators of order −n and type zero in Boutet de Monvel's calculus form a subset of Dixmier's trace ideal for the Hilbert space of L
2 sections in vector bundles E over X, F over ∂X.
We show that, on these operators, Dixmier's trace can be computed in terms of the same expressions that determine the noncommutative
residue. In particular it is independent of the averaging procedure. However, the noncommutative residue and Dixmier's trace
are not multiples of each other unless the boundary is empty.
As a corollary we show how to compute Dixmier's trace for parametrices or inverses of classical elliptic boundary value
problems of the form Pu=f; Tu=0 with an elliptic differential operator P of order n in the interior and a trace operator T. In this particular situation, Dixmier's trace and the noncommutative residue do coincide up to a factor.
Received: Received: 13 January 1998 相似文献
2.
Violeta Petkova 《Integral Equations and Operator Theory》2007,59(3):355-378
A Wiener–Hopf operator on a Banach space of functions on is a bounded operator T such that P
+
S
−a
TS
a
= T, a ≥ 0, where S
a
is the operator of translation by a. We obtain a representation theorem for the Wiener–Hopf operators on a large class of functions on with values in a separable Hilbert space.
相似文献
3.
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
4.
In 1991 Effros and Ruan conjectured that a certain Grothendieck-type inequality for a bilinear form on C*-algebras holds if (and only if) the bilinear form is jointly completely bounded. In 2002 Pisier and Shlyakhtenko proved that
this inequality holds in the more general setting of operator spaces, provided that the operator spaces in question are exact.
Moreover, they proved that the conjecture of Effros and Ruan holds for pairs of C*-algebras, of which at least one is exact. In this paper we prove that the Effros–Ruan conjecture holds for general C*-algebras, with constant one. More precisely, we show that for every jointly completely bounded (for short, j.c.b.) bilinear
form on a pair of C*-algebras A and B, there exist states f
1, f
2 on A and g
1, g
2 on B such that for all a∈A and b∈B,
While the approach by Pisier and Shlyakhtenko relies on free probability techniques, our proof uses more classical operator
algebra theory, namely, Tomita–Takesaki theory and special properties of the Powers factors of type IIIλ, 0<λ<1.
Mathematics Subject Classification (2000) 46L10, 47L25 相似文献
5.
Mahmoud Baroun Lahcen Maniar Roland Schnaubelt 《Integral Equations and Operator Theory》2009,65(2):169-193
We show the existence and uniqueness of the (asymptotically) almost periodic solution to parabolic evolution equations with
inhomogeneous boundary values on
\mathbbR{\mathbb{R}} and
\mathbbR±\mathbb{R}_{\pm}, if the data are (asymptotically) almost periodic. We assume that the underlying homogeneous problem satisfies the ‘Acquistapace–Terreni’
conditions and has an exponential dichotomy. If there is an exponential dichotomy only on half intervals ( − ∞, − T] and [T, ∞), then we obtain a Fredholm alternative of the equation on
\mathbbR{\mathbb{R}} in the space of functions being asymptotically almost periodic on
\mathbbR+{\mathbb{R}}_{+} and
\mathbbR-\mathbb{R}_{-}. 相似文献
6.
S. Mecheri 《Czechoslovak Mathematical Journal》2007,57(2):697-703
Let ℋ be a separable infinite dimensional complex Hilbert space, and let ℒ(H) denote the algebra of all bounded linear operators on ℋ into itself. Let A = (A
1, A
2,..., A
n), B = (B
1, B
2,..., B
n) be n-tuples of operators in ℒ(H); we define the elementary operators Δ
A,B
: ℒ(H) ↦ ℒ(H) by
.
In this paper, we characterize the class of pairs of operators A, B ∈ ℒ(H) satisfying Putnam-Fuglede’s property, i.e, the class of pairs of operators A,B ∈ ℒ(H) such that
implies
for all T ∈ C
1 (H) (trace class operators). The main result is the equivalence between this property and the fact that the ultraweak closure
of the range of the elementary operator ΔA,B is closed under taking adjoints. This leads us to give a new characterization of the orthogonality (in the sense of Birkhoff)
of the range of an elementary operator and its kernel in C
1 classes.
This work was supported by the research center project No. 2005-04. 相似文献
7.
Suppose T+(E){\mathcal{T}_{+}(E)} is the tensor algebra of a W*-correspondence E and H
∞(E) is the associated Hardy algebra. We investigate the problem of extending completely contractive representations of T+(E){\mathcal{T}_{+}(E)} on a Hilbert space to ultra-weakly continuous completely contractive representations of H
∞(E) on the same Hilbert space. Our work extends the classical Sz.-Nagy–Foiaş functional calculus and more recent work by Davidson,
Li and Pitts on the representation theory of Popescu’s noncommutative disc algebra. 相似文献
8.
Julián Fernández Bonder Nicolas Saintier 《Annali di Matematica Pura ed Applicata》2008,187(4):683-704
In this paper we find estimates for the optimal constant in the critical Sobolev trace inequality that are independent of Ω. This estimates generalized those of Adimurthi and Yadava (Comm Partial Diff Equ 16(11):1733–1760,
1991) for general p. Here p
* : = p(N − 1)/(N − p) is the critical exponent for the immersion and N is the space dimension. Then we apply our results first to prove existence of positive solutions to a nonlinear elliptic
problem with a nonlinear boundary condition with critical growth on the boundary, generalizing the results of Fernández Bonder
and Rossi (Bull Lond Math Soc 37:119–125, 2005). Finally, we study an optimal design problem with critical exponent.
相似文献
9.
Isabelle Chalendar Antoine Flattot Nadine Guillotin-Plantard 《Archiv der Mathematik》2008,90(4):353-359
In this paper we study the point spectrum of the operator
where d ≥ 1, 1 ≤ p ≤ ∞([0, 1]
d
) and τ is an irrational rotation on [0, 1]
d
. For a particular class of weights w, the point spectrum of T
w,τ is shown to be empty, generalizing Davie’s result [3], who considered the case p = 2, d = 1.
Received: 1 June 2007, Revised: 16 October 2007 相似文献
10.
Roberta Tognari 《Potential Analysis》2007,26(2):163-188
We consider the operator in L
2(B, ν) and in L
1(B, ν) with Neumann boundary condition, where U is an unbounded function belonging to for some q ∈(1, ∞), B is the possibly unbounded convex open set in where U is finite and ν(dx) = C exp (−2U (x))dx is a probability measure, infinitesimally invariant for N
0. We prove that the closure of N
0 is a m-dissipative operator both in L
2(B, ν) and in L
1(B, ν). Moreover we study the properties of ergodicity and strong mixing of the measure ν in the L
2 case.
相似文献
11.
For uniformly stable bounded analytic C
0-semigroups {T(t)}
t≥0 of linear operators in a Banach space B, we study the behavior of their orbits T (t)x, x ∈ B, at infinity. We also analyze the relationship between the order of approaching the orbit T (t)x to zero as t → ∞ and the degree of smoothness of the vector x with respect to the operator A
−1 inverse to the generator A of the semigroup {T(t)}
t≥0. In particular, it is shown that, for this semigroup, there exist orbits approaching zero at infinity not slower than
, where a > 0, 0 < α < π/(2(π-θ)), θ is the angle of analyticity of {T(t)}
t≥0, and the collection of these orbits is dense in the set of all orbits.
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 2, pp. 148–159, February, 2006. 相似文献
12.
Assume that each completely irrational noncommutative torus is realized as an inductive limit of circle algebras, and that for a completely irrational noncommutative torus Aw of rank m there are a completely irrational noncommutative torus Aρ of rank m and a positive integer d such that tr(Aw)=1/d.tr(Aρ).It is proved that the set of all C^*-algebras of sections of locally trivial C^*-algebra bundles over S^2 with fibres Aω has a group sturcture,denoted by π1^s(Aut(Aω)),which is isomorphic to Zif Ed>1 and {0} if d>1.Let Bcd be a cd-homogeneous C^*-algebra over S^2×T^2 of which no non-trivial matrix algebra can be factored out.The spherical noncommutative torus Sρ^cd is defined by twisting C^*(T2×Z^m-2) in Bcd ×C^*(Z^m-3) by a totally skew multiplier ρ on T^2×Z^m-2。It is shown that Sρ^cd×Mρ∞ is isomorphic to C(S^2)×C^*(T^2×Z^m-2,ρ)× Mcd(C)×Mρ∞ if and only if the set of prime factors of cd is a subset of the set of prime factors of p. 相似文献
13.
Let {φ k } be an orthonormal system on a quasi-metric measure space ${\mathbb{X}}Let {φ
k
} be an orthonormal system on a quasi-metric measure space
\mathbbX{\mathbb{X}}, {ℓ
k
} be a nondecreasing sequence of numbers with lim
k→∞
ℓ
k
=∞. A diffusion polynomial of degree L is an element of the span of {φ
k
:ℓ
k
≤L}. The heat kernel is defined formally by Kt(x,y)=?k=0¥exp(-lk2t)fk(x)[`(fk(y))]K_{t}(x,y)=\sum_{k=0}^{\infty}\exp(-\ell _{k}^{2}t)\phi_{k}(x)\overline{\phi_{k}(y)}. If T is a (differential) operator, and both K
t
and T
y
K
t
have Gaussian upper bounds, we prove the Bernstein inequality: for every p, 1≤p≤∞ and diffusion polynomial P of degree L, ‖TP‖
p
≤c
1
L
c
‖P‖
p
. In particular, we are interested in the case when
\mathbbX{\mathbb{X}} is a Riemannian manifold, T is a derivative operator, and p 1 2p\not=2. In the case when
\mathbbX{\mathbb{X}} is a compact Riemannian manifold without boundary and the measure is finite, we use the Bernstein inequality to prove the
existence of quadrature formulas exact for integrating diffusion polynomials, based on an arbitrary data. The degree of the diffusion polynomials for which this formula is exact depends upon the mesh norm of the data. The
results are stated in greater generality. In particular, when T is the identity operator, we recover the earlier results of Maggioni and Mhaskar on the summability of certain diffusion
polynomial valued operators. 相似文献
14.
Let T and be arbitrary nonnegative, irreducible, stochastic matrices corresponding to two ergodic Markov chains on n states. A function κ is called a condition number for Markov chains with respect to the (α, β)–norm pair if . Here π and are the stationary distribution vectors of the two chains, respectively. Various condition numbers, particularly with respect
to the (1, ∞) and (∞, ∞)-norm pairs have been suggested in the literature. They were ranked according to their size by Cho
and Meyer in a paper from 2001. In this paper we first of all show that what we call the generalized ergodicity coefficient
, where e is the n-vector of all 1’s and A
# is the group generalized inverse of A = I − T, is the smallest condition number of Markov chains with respect to the (p, ∞)-norm pair. We use this result to identify the smallest condition number of Markov chains among the (∞, ∞) and (1, ∞)-norm
pairs. These are, respectively, κ
3 and κ
6 in the Cho–Meyer list of 8 condition numbers. Kirkland has studied κ
3(T). He has shown that and he has characterized transition matrices for which equality holds. We prove here again that 2κ
3(T) ≤ κ(6) which appears in the Cho–Meyer paper and we characterize the transition matrices T for which . There is actually only one such matrix: T = (J
n
− I)/(n − 1), where J
n
is the n × n matrix of all 1’s.
This research was supported in part by NSERC under Grant OGP0138251 and NSA Grant No. 06G–232. 相似文献
15.
Mustapha Mokhtar-Kharroubi 《Journal of Evolution Equations》2008,8(2):327-352
We deal with streaming operators T
H
defined in L
1 spaces by the directional derivative with positive boundary operator H of norm 1 relating the incoming and outgoing fluxes. It is known that T
H
need not be a generator but there exists a contraction semigroup generated by an extension A of T
H
. This paper deals with the total mass carried by individual trajectories {e
tA
f; t ≥ 0} for nonnegative initial data f and related topics. In particular, our analysis covers the problem of (the lack of) stochasticity of {e
tA
; t ≥ 0} for conservative boundary operator H.
相似文献
16.
Let ℳ be a von Neumann factor of type II1 with a normalized trace τ. In 1983 L. G. Brown showed that to every operator T∈ℳ one can in a natural way associate a spectral distribution measure
μ
T (now called the Brown measure of T), which is a probability measure in ℂ with support in the spectrum σ(T) of T. In this paper it is shown that for every T∈ℳ and every Borel set B in ℂ, there is a unique closed T-invariant subspace
affiliated with ℳ, such that the Brown measure of
is concentrated on B and the Brown measure of
is concentrated on ℂ∖B. Moreover,
is T-hyperinvariant and the trace of
is equal to μ
T(B). In particular, if T∈ℳ has a Brown measure which is not concentrated on a singleton, then there exists a non-trivial, closed,
T-hyperinvariant subspace. Furthermore, it is shown that for every T∈ℳ the limit
exists in the strong operator topology, and the projection onto
is equal to 1[0,r](A), for every r>0.
Supported by The Danish National Research Foundation. 相似文献
17.
General results of interpolation (e.g., Nevanlinna-Pick) by elements in the noncommutative analytic Toeplitz algebraF
∞ (resp., noncommutative disc algebraA
n) with consequences to the interpolation by bounded operator-valued analytic functions in the unit ball of ℂn are obtained. Noncommutative Poisson transforms are used to provide new von Neumann type inequalities. Completely isometric
representations of the quotient algebraF
∞/J on Hilbert spaces whereJ is anyw
*-closed, 2-sided ideal ofF
∞, are obtained and used to construct aw
*-continuous,F
∞/J-functional calculus associated to row contractionsT=[T
1,…,T
n] whenf(T1, …, Tn)=0 for anyf∈J. Other properties of the dual algebraF
∞/J are considered.
The second author was partially supported by NSF DMS-9531954. 相似文献
18.
We prove weighted strong inequalities for the multilinear potential operator Tf{\cal T}_{\phi} and its commutator, where the kernel ϕ satisfies certain growth condition. For these operators we also obtain Fefferman-Stein type inequalities and Coifman type
estimates. Moreover we prove weighted weak type inequalities for the multilinear maximal operator Mj,LB\mathcal{M}_{\varphi,L^{B}} associated to an essentially nondecreasing function φ and to the Orlicz space L
B
for a given Young function B. This result allows us to obtain a weighted weak type inequality for the operator Tf{\cal T}_{\phi}. 相似文献
19.
Let X be a Banach space and suppose that A
1,…,A
n
are noncommuting (that is, not necessarily commuting) elements in ℒ(X), the space of bounded linear operators on X. Further, for each i∈{1,…,n}, let μ
i
be a continuous probability measure on ℬ([0,1]), the Borel class of [0,1]. Each such n-tuple of operator-measure pairs (A
i
,μ
i
), i=1,…,n, determines an operational calculus or disentangling map
Tm1,...,mn{\mathcal{T}}_{\mu_{1},\dots,\mu_{n}}
from a commutative Banach algebra
\mathbbD(A1,...,An){\mathbb{D}}(A_{1},\dots,A_{n})
of analytic functions, called the disentangling algebra , into the noncommutative Banach algebra ℒ(X). The disentanglings are the central processes of Feynman’s operational calculi. 相似文献
20.
Let B = (B
1(t), . . . , B
d
(t)) be a d-dimensional fractional Brownian motion with Hurst index α ≤ 1/4, or more generally a Gaussian process whose paths have the same local regularity. Defining properly iterated integrals
of B is a difficult task because of the low H?lder regularity index of its paths. Yet rough path theory shows it is the key to
the construction of a stochastic calculus with respect to B, or to solving differential equations driven by B. We intend to show in a forthcoming series of papers how to desingularize iterated integrals by a weak singular non-Gaussian
perturbation of the Gaussian measure defined by a limit in law procedure. Convergence is proved by using “standard” tools
of constructive field theory, in particular cluster expansions and renormalization. These powerful tools allow optimal estimates
of the moments and call for an extension of the Gaussian tools such as for instance the Malliavin calculus. This first paper
aims to be both a presentation of the basics of rough path theory to physicists, and of perturbative field theory to probabilists;
it is only heuristic, in particular because the desingularization of iterated integrals is really a non-perturbative effect. It is also meant to be a general motivating introduction to the subject, with some insights into quantum field theory
and stochastic calculus. The interested reader should read for a second time the companion article (Magnen and Unterberger
in From constructive theory to fractional stochastic calculus. (II) The rough path for
\frac16 < a < \frac14{\frac{1}{6} < \alpha < \frac{1}{4}}: constructive proof of convergence, 2011, preprint) for the constructive proofs. 相似文献