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1.
Let G be an additive subgroup of a normed space X. We say that a point is weakly separated (resp. -separated) from G if it can be separated from G by a continuous character (resp. by a continuous positive definite function). Let T : X → Y be a continuous linear operator. Consider the following conditions:
(ws) if , then x is weakly separated from G;
(ps) if , then x is -separated from G;
(wp) if Tx is -separated from T(G), then x is weakly separated from G.
By (resp. , ) we denote the class of operators T : X → Y which satisfy (ws) (resp. (ps), (wp)) for all and all subgroups G of X. The paper is an attempt to describe the above classes of operators for various Banach spaces X, Y. It is proved that if X, Y are Hilbert spaces, then is the class of Hilbert-Schmidt operators. It is also shown that if T is a Hilbert-to-Banach space operator with finite ℓ-norm, then .
相似文献
2.
Raúl E. Curto Lawrence A. Fialkow H. Michael Möller 《Integral Equations and Operator Theory》2008,60(2):177-200
For a degree 2n real d-dimensional multisequence to have a representing measure μ, it is necessary for the associated moment matrix to be positive semidefinite and for the algebraic variety associated to β, , to satisfy rank card as well as the following consistency condition: if a polynomial vanishes on , then . We prove that for the extremal case , positivity of and consistency are sufficient for the existence of a (unique, rank -atomic) representing measure. We also show that in the preceding result, consistency cannot always be replaced by recursiveness
of .
The first-named author’s research was partially supported by NSF Research Grants DMS-0099357 and DMS-0400741. The second-named
author’s research was partially supported by NSF Research Grant DMS-0201430 and DMS-0457138. 相似文献
3.
Gelu Popescu 《Mathematische Annalen》2008,342(1):1-30
In this paper we obtain a noncommutative multivariable analogue of the classical Nevanlinna–Pick interpolation problem for
analytic functions with positive real parts on the open unit disc. Given a function , where is an arbitrary subset of the open unit ball , we find necessary and sufficient conditions for the existence of a free holomorphic function g with complex coefficients on the noncommutative open unit ball such that
where is the algebra of all bounded linear operators on a Hilbert space . The proof employs several results from noncommutative multivariable operator theory and a noncommutative Cayley transform
(introduced and studied in the present paper) acting from the set of all free holomorphic functions with positive real parts
to the set of all bounded free holomorphic functions. All the results of this paper are obtained in the more general setting
of free holomorphic functions with operator-valued coefficients. As consequences, we deduce some results concerning operator-valued
analytic interpolation on the unit ball .
Research supported in part by an NSF grant. 相似文献
4.
Yucheng Li 《Integral Equations and Operator Theory》2009,63(1):95-102
Let D be the unit disk and be the weighted Bergman space. In this paper, we prove that the multiplication operator is similar to
M
z
on .
The author was supported in part by NSF Grant (10571041, L2007B05). 相似文献
5.
M. A. Bastos C. A. Fernandes Yu. I. Karlovich 《Integral Equations and Operator Theory》2006,55(1):19-67
We establish a symbol calculus for the C*-subalgebra
of
generated by the operators of multiplication by slowly oscillating and piecewise continuous functions and the operators
where
is the Cauchy singular integral operator and
The C*-algebra
is invariant under the transformations
where Uz is the rotation operator
Using the localtrajectory method, which is a natural generalization of the Allan-Douglas local principle to nonlocal type
operators, we construct symbol calculi and establish Fredholm criteria for the C*-algebra
generated by the operators
and
for the C*-algebra
generated by the operators
and
and for the C*-algebra
generated by the algebras
and
The C*-algebra
can be considered as an algebra of convolution type operators with piecewise slowly oscillating coefficients and shifts acting
freely. 相似文献
6.
Let denote the set of simultaneously - approximable points in and denote the set of multiplicatively ψ-approximable points in . Let be a manifold in . The aim is to develop a metric theory for the sets and analogous to the classical theory in which is simply . In this note, we mainly restrict our attention to the case that is a planar curve . A complete Hausdorff dimension theory is established for the sets and . A divergent Khintchine type result is obtained for ; i.e. if a certain sum diverges then the one-dimensional Lebesgue measure on of is full. Furthermore, in the case that is a rational quadric the convergent Khintchine type result is obtained for both types of approximation. Our results for
naturally generalize the dimension and Lebesgue measure statements of Beresnevich et al. (Mem AMS, 179 (846), 1–91 (2006)). Moreover, within the multiplicative framework, our results for constitute the first of their type.
The research of Victor V. Beresnevich was supported by an EPSRC Grant R90727/01. Sanju L. Velani is a Royal Society University
Research Fellow.
For Iona and Ayesha on No. 3. 相似文献
7.
The Aronszajn–Donoghue Theory for Rank One Perturbations of the
$$\mathcal{H}_{-2} {\text{-Class}}$$
A singular rank one perturbation
of a self-adjoint operator A in a Hilbert space
is considered, where
and
but
with
the usual A–scale of Hilbert spaces. A modified version of the Aronszajn-Krein formula is given. It has the form
where F denotes the regularized Borel transform of the scalar spectral measure of A associated with . Using this formula we develop a variant of the well known Aronszajn–Donoghue spectral theory for a general rank one perturbation of the
class.Submitted: March 14, 2002 Revised: December 15, 2002 相似文献
8.
Alexander Prestel 《manuscripta mathematica》2007,123(1):95-103
Let K be an algebraically closed field with a valuation ring or a real closed field with a convex valuation ring . We show that the projection of a basic (see “Introduction”) subset of to K
n
is again basic. 相似文献
9.
Jaume Llibre Clàudia Valls 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2009,15(2):237-245
We study the analytical integrability of the FitzHugh–Nagumo systems in with parameters
相似文献
10.
For a class of essentially normal operators, we characterize their norm closures of
–orbits. Moreover, we introduce a notion of the quasiapproximate
– equivalence of essentially normal operators and determine completely the quasiapproximate
–invariants. Finally, we give the canonical forms of essentially normal operators under this quasiapproximate
–equivalence. 相似文献
11.
M. Amélia Bastos Claudio A. Fernandes Yuri I. Karlovich 《Complex Analysis and Operator Theory》2008,2(2):241-272
The C*-subalgebra of generated by all multiplication operators by slowly oscillating and piecewise continuous functions, by the Cauchy singular
integral operator and by the range of a unitary representation of an amenable group of diffeomorphisms with any nonempty set of common fixed points is studied. A symbol calculus for the C*-algebra and a Fredholm criterion for its elements are obtained. For the C*-algebra composed by all functional operators in , an invertibility criterion for its elements is also established. Both the C*-algebras and are investigated by using a generalization of the local-trajectory method for C*-algebras associated with C*-dynamical systems which is based on the notion of spectral measure.
Submitted: April 30, 2007. Accepted: November 5, 2007. 相似文献
12.
Egor A. Alekhno 《Positivity》2009,13(1):3-20
Let T be a positive operator on a Banach lattice E. Some properties of Weyl essential spectrum σew(T), in particular, the equality , where is the set of all compact operators on E, are established. If r(T) does not belong to Fredholm essential spectrum σef(T), then for every a ≠ 0, where T−1 is a residue of the resolvent R(., T) at r(T). The new conditions for which implies , are derived. The question when the relation holds, where is Lozanovsky’s essential spectrum, will be considered. Lozanovsky’s order essential spectrum is introduced. A number of
auxiliary results are proved. Among them the following generalization of Nikol’sky’s theorem: if T is an operator of index zero, then T = R + K, where R is invertible, K ≥ 0 is of finite rank. Under the natural assumptions (one of them is ) a theorem about the Frobenius normal form is proved: there exist T-invariant bands such that if
, where , then an operator on Di is band irreducible.
相似文献
13.
Let and be C*-dynamical systems and assume that is a separable simple C*-algebra and that α and β are *-automorphisms. Then the semicrossed products and are isometrically isomorphic if and only if the dynamical systems and are outer conjugate.
K. R. Davidson was partially supported by an NSERC grant. E. G. Katsoulis was partially supported by a summer grant from ECU 相似文献
14.
15.
New variational principles based on the concept of anti-selfdual (ASD) Lagrangians were recently introduced in “AIHP-Analyse
non linéaire, 2006”. We continue here the program of using such Lagrangians to provide variational formulations and resolutions
to various basic equations and evolutions which do not normally fit in the Euler-Lagrange framework. In particular, we consider
stationary boundary value problems of the form as well ass dissipative initial value evolutions of the form where is a convex potential on an infinite dimensional space, A is a linear operator and is any scalar. The framework developed in the above mentioned paper reformulates these problems as and respectively, where is an “ASD” vector field derived from a suitable Lagrangian L. In this paper, we extend the domain of application of this approach by establishing existence and regularity results under
much less restrictive boundedness conditions on the anti-selfdual Lagrangian L so as to cover equations involving unbounded operators. Our main applications deal with various nonlinear boundary value
problems and parabolic initial value equations governed by transport operators with or without a diffusion term.
Nassif Ghoussoub research was partially supported by a grant from the Natural Sciences and Engineering Research Council of
Canada. The author gratefully acknowledges the hospitality and support of the Centre de Recherches Mathématiques in Montréal
where this work was initiated.
Leo Tzou’s research was partially supported by a doctoral postgraduate scholarship from the Natural Science and Engineering
Research Council of Canada. 相似文献
16.
James Gillespie 《Mathematische Zeitschrift》2007,257(4):811-843
We put a monoidal model category structure on the category of chain complexes of quasi-coherent sheaves over a quasi-compact
and semi-separated scheme X. The approach generalizes and simplifies the method used by the author in (Trans Am Math Soc 356(8) 3369–3390, 2004) and
(Trans Am Math Soc 358(7), 2855–2874, 2006) to build monoidal model structures on the category of chain complexes of modules
over a ring and chain complexes of sheaves over a ringed space. Indeed, much of the paper is dedicated to showing that in
any Grothendieck category , any nice enough class of objects induces a model structure on the category Ch() of chain complexes. The main technical requirement on is the existence of a regular cardinal κ such that every object satisfies the following property: Each κ-generated subobject of F is contained in another κ-generated subobject S for which . Such a class is called a Kaplansky class. Kaplansky classes first appeared in Enochs and López-Ramos (Rend Sem Mat Univ Padova 107, 67–79,
2002) in the context of modules over a ring R. We study in detail the connection between Kaplansky classes and model categories. We also find simple conditions to put
on which will guarantee that our model structure is monoidal. We will see that in several categories the class of flat objects
form such Kaplansky classes, and hence induce monoidal model structures on the associated chain complex categories. We will
also see that in any Grothendieck category , the class of all objects is a Kaplansky class which induces the usual (non-monoidal) injective model structure on Ch(). 相似文献
17.
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
18.
Emanuele Delucchi 《Journal of Algebraic Combinatorics》2007,26(4):477-494
Given a finite group G and a natural number n, we study the structure of the complex of nested sets of the associated Dowling lattice
(Proc. Internat. Sympos., 1971, pp. 101–115) and of its subposet of the G-symmetric partitions
which was recently introduced by Hultman (, 2006), together with the complex of G-symmetric phylogenetic trees
. Hultman shows that the complexes
and
are homotopy equivalent and Cohen–Macaulay, and determines the rank of their top homology.
An application of the theory of building sets and nested set complexes by Feichtner and Kozlov (Selecta Math. (N.S.)
10, 37–60, 2004) shows that in fact
is subdivided by the order complex of
. We introduce the complex of Dowling trees
and prove that it is subdivided by the order complex of
. Application of a theorem of Feichtner and Sturmfels (Port. Math. (N.S.)
62, 437–468, 2005) shows that, as a simplicial complex,
is in fact isomorphic to the Bergman complex of the associated Dowling geometry.
Topologically, we prove that
is obtained from
by successive coning over certain subcomplexes. It is well known that
is shellable, and of the same dimension as
. We explicitly and independently calculate how many homology spheres are added in passing from
to
. Comparison with work of Gottlieb and Wachs (Adv. Appl. Math.
24(4), 301–336, 2000) shows that
is intimely related to the representation theory of the top homology of
.
Research partially supported by the Swiss National Science Foundation, project PP002-106403/1. 相似文献
19.
Let be a convex function and be its Legendre tranform. It is proved that if is invariant by changes of signs, then . This is a functional version of the inverse Santaló inequality for unconditional convex bodies due to J. Saint Raymond.
The proof involves a general result on increasing functions on together with a functional form of Lozanovskii’s lemma. In the last section, we prove that for some c > 0, one has always . This generalizes a result of B. Klartag and V. Milman.
相似文献
20.
We study cyclicity of operators on a separable Banach space which admit a bicyclic vector such that the norms of its images
under the iterates of the operator satisfy certain growth conditions. A simple consequence of our main result is that a bicyclic
unitary operator on a Banach space with separable dual is cyclic. Our results also imply that if is the shift operator acting on the weighted space of sequences , if the weight ω satisfies some regularity conditions and ω(n) = 1 for nonnegative n, then S is cyclic if . On the other hand one can see that S is not cyclic if the series diverges. We show that the question of Herrero whether either S or S* is cyclic on admits a positive answer when the series is convergent. We also prove completeness results for translates in certain Banach spaces of functions on . 相似文献