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1.
Let M be a II-factor and denote by τ its normal faithful semi-finite trace. For any rearrangement invariant Köthe function space X on [0,+∞[, let X(M,τ) be the associated non-commutative Banach function space. This paper is concerned with ideals in M of the form IX(M,τ)=MX(M,τ) that are contained in Lp(M,τ) for some p>0. It is proved that an element T in IX(M,τ) is a finite sum of commutators of the form [A,B] with AIX(M,τ) and BM if and only if the function belongs to X, where νT is the Brown spectral measure of T and tλt(T) is the non-increasing rearrangement of the function λ→|λ| with respect to νT. This extends to general Banach function spaces a result obtained by Kalton for quasi-Banach ideals of compact operators and implies that the Dixmier's trace of a quasi-nilpotent element in L1,∞(M,τ) is always zero.  相似文献   

2.
Suppose A is a dual Banach algebra, and a representation π:AB(?2) is unital, weak* continuous, and contractive. We use a “Hilbert-Schmidt version” of Arveson distance formula to construct an operator space X, isometric to ?2⊗?2, such that the space of completely bounded maps on X consists of Hilbert-Schmidt perturbations of π(A)⊗I?2. This allows us to establish the existence of operator spaces with various interesting properties. For instance, we construct an operator space X for which the group K1(CB(X)) contains Z2 as a subgroup, and a completely indecomposable operator space containing an infinite dimensional homogeneous Hilbertian subspace.  相似文献   

3.
For all positive a the point spectrum of the (C, α) matrix is determined, where the matrix is regarded as an operator on certain Banach sequence spaces. In particular the point spectrum is obtained in the spaces Ip(X), with 1<p≤∞, where X is a Banach space.  相似文献   

4.
Let X and Y be given Banach spaces. For AB(X), BB(Y) and CB(Y,X), let MC be the operator defined on XY by . In this paper we give conditions for continuity of τ at MC through continuity of τ at A and B, where τ can be equal to the spectrum or approximate point spectrum.  相似文献   

5.
Let (X,τ) be a topological space and let ρ be a metric defined on X. We shall say that (X,τ) is fragmented by ρ if whenever ε>0 and A is a nonempty subset of X there is a τ-open set U such that UA≠∅ and ρ−diam(UA)<ε. In this paper we consider the notion of fragmentability, and its generalisation σ-fragmentability, in the setting of topological groups and metric-valued function spaces. We show that in the presence of Baireness fragmentability of a topological group is very close to metrizability of that group. We also show that for a compact Hausdorff space X, σ-fragmentability of (C(X),‖⋅) implies that the space Cp(X;M) of all continuous functions from X into a metric space M, endowed with the topology of pointwise convergence on X, is fragmented by a metric whose topology is at least as strong as the uniform topology on C(X;M). The primary tool used is that of topological games.  相似文献   

6.
Let B(X) be the algebra of all bounded linear operators on the Banach space X, and let N(X) be the set of nilpotent operators in B(X). Suppose ?:B(X)→B(X) is a surjective map such that A,BB(X) satisfy ABN(X) if and only if ?(A)?(B)∈N(X). If X is infinite dimensional, then there exists a map f:B(X)→C?{0} such that one of the following holds:
(a)
There is a bijective bounded linear or conjugate-linear operator S:XX such that ? has the form A?S[f(A)A]S-1.
(b)
The space X is reflexive, and there exists a bijective bounded linear or conjugate-linear operator S : X′ → X such that ? has the form A ? S[f(A)A′]S−1.
If X has dimension n with 3 ? n < ∞, and B(X) is identified with the algebra Mn of n × n complex matrices, then there exist a map f:MnC?{0}, a field automorphism ξ:CC, and an invertible S ∈ Mn such that ? has one of the following forms:
  相似文献   

7.
Let X be a nonempty, convex and compact subset of normed linear space E (respectively, let X be a nonempty, bounded, closed and convex subset of Banach space E and A be a nonempty, convex and compact subset of X) and f:X×XR be a given function, the uniqueness of equilibrium point for equilibrium problem which is to find xX (respectively, xA) such that f(x,y)≥0 for all yX (respectively, f(x,y)≥0 for all yA) is studied with varying f (respectively, with both varying f and varying A). The results show that most of equilibrium problems (in the sense of Baire category) have unique equilibrium point.  相似文献   

8.
It is known that the algebra of Schur operators on ?2 (namely operators bounded on both ?1 and ?) is not inverse-closed. When ?2=?2(X) where X is a metric space, one can consider elements of the Schur algebra with certain decay at infinity. For instance if X has the doubling property, then Q. Sun has proved that the weighted Schur algebra Aω(X) for a strictly polynomial weight ω is inverse-closed. In this paper, we prove a sharp result on left-invertibility of the these operators. Namely, if an operator AAω(X) satisfies ‖Afp?‖fp, for some 1?p?∞, then it admits a left-inverse in Aω(X). The main difficulty here is to obtain the above inequality in ?2. The author was both motivated and inspired by a previous work of Aldroubi, Baskarov and Krishtal (2008) [1], where similar results were obtained through different methods for X=Zd, under additional conditions on the decay.  相似文献   

9.
Let us consider two nonempty subsets A,B of a normed linear space X, and let us denote by 2B the set of all subsets of B. We introduce a new class of multivalued mappings {T:A→2B}, called R-KKM mappings, which extends the notion of KKM mappings. First, we discuss some sufficient conditions for which the set ∩{T(x):xA} is nonempty. Using this nonempty intersection theorem, we attempt to prove a extended version of the Fan-Browder multivalued fixed point theorem, in a normed linear space setting, by providing an existence of a best proximity point.  相似文献   

10.
Suppose (B,β) is an operator ideal, and A is a linear space of operators between Banach spaces X and Y. Modifying the classical notion of hyperreflexivity, we say that A is called B-hyperreflexive if there exists a constant C such that, for any TB(X,Y) with α=supβ(qTi)<∞ (the supremum runs over all isometric embeddings i into X, and all quotient maps of Y, satisfying qAi=0), there exists aA, for which β(Ta)?Cα. In this paper, we give examples of B-hyperreflexive spaces, as well as of spaces failing this property. In the last section, we apply SE-hyperreflexivity of operator algebras (SE is a regular symmetrically normed operator ideal) to constructing operator spaces with prescribed families of completely bounded maps.  相似文献   

11.
It is shown that if A, B, X are Hilbert space operators such that X?γI, for the positive real number γ, and p,q>1 with 1/p+1/q=1, then |AB|2?p|A|2+q|B|2 with equality if and only if (1−p)A=B and γ||||AB|2|||?|||p|A|2X+qX|B|2||| for every unitarily invariant norm. Moreover, if in addition A, B are normal and X is any Hilbert-Schmidt operator, then ‖δA,B2(X)‖2?‖p|A|2X+qX|B|22 with equality if and only if (1−p)AX=XB.  相似文献   

12.
We investigate some subtle and interesting phenomena in the duality theory of operator spaces and operator algebras, and give several applications of the surprising fact that certain maps are always weak*-continuous on dual spaces. In particular, if X is a subspace of a C*-algebra A, and if aA satisfies aXX, then we show that the function x?ax on X is automatically weak* continuous if either (a) X is a dual operator space, or (b) a*XX and X is a dual Banach space. These results hinge on a generalization to Banach modules of Tomiyama's famous theorem on contractive projections onto a C*-subalgebra. Applications include a new characterization of the σ-weakly closed (possibly nonunital and nonselfadjoint) operator algebras, and a generalization of the theory of W*-modules to the framework of modules over such algebras. We also give a Banach module characterization of σ-weakly closed spaces of operators which are invariant under the action of a von Neumann algebra.  相似文献   

13.
Let A and B be (not necessarily unital or closed) standard operator algebras on complex Banach spaces X and Y, respectively. For a bounded linear operator A on X, the peripheral spectrum σπ(A) of A is the set σπ(A)={zσ(A):|z|=maxωσ(A)|ω|}, where σ(A) denotes the spectrum of A. Assume that Φ:AB is a map the range of which contains all operators of rank at most two. It is shown that the map Φ satisfies the condition that σπ(BAB)=σπ(Φ(B)Φ(A)Φ(B)) for all A,BA if and only if there exists a scalar λC with λ3=1 and either there exists an invertible operator TB(X,Y) such that Φ(A)=λTAT-1 for every AA; or there exists an invertible operator TB(X,Y) such that Φ(A)=λTAT-1 for every AA. If X=H and Y=K are complex Hilbert spaces, the maps preserving the peripheral spectrum of the Jordan skew semi-triple product BAB are also characterized. Such maps are of the form A?UAU or A?UAtU, where UB(H,K) is a unitary operator, At denotes the transpose of A in an arbitrary but fixed orthonormal basis of H.  相似文献   

14.
It is shown that if X is a countably paracompact collectionwise normal space, Y is a Banach space and φ:XY2 is a lower semicontinuous mapping such that φ(x) is Y or a compact convex subset with Cardφ(x)>1 for each xX, then φ admits a continuous selection f:XY such that f(x) is not an extreme point of φ(x) for each xX. This is an affirmative answer to the problem posed by V. Gutev, H. Ohta and K. Yamazaki [V. Gutev, H. Ohta and K. Yamazaki, Selections and sandwich-like properties via semi-continuous Banach-valued functions, J. Math. Soc. Japan 55 (2003) 499-521].  相似文献   

15.
A Hilbert bundle (p, B, X) is a type of fibre space p:BX such that each fibre p?1(x) is a Hilbert space. However, p?1(x) may vary in dimension as x varies in X. We generalize the classical homotopy classification theory of vector bundles to a “homotopy” classification of certain Hilbert bundles. An (m, n)-bundle over the pair (X, A) is a Hilbert bundle (p, B, X) such that the dimension of p?1(x) is m for x in A and n otherwise. The main result here is that if A is a compact set lying in the “edge” of the metric space X (e.g. if X is a topological manifold and A is a compact subset of the boundary of X), then the problem of classifying (m, n)-bundles over (X, A) reduces to a problem in the classical theory of vector bundles. In particular, we show there is a one-to-one correspondence between the members of the orbit set, [A, Gm(Cn)]/[X, U(n)] ¦ A, and the isomorphism classes of (m, n)-bundles over (X, A) which are trivial over X, A.  相似文献   

16.
Given a space M, a family of sets A of a space X is ordered by M if A={AK:K is a compact subset of M} and KL implies AKAL. We study the class M of spaces which have compact covers ordered by a second countable space. We prove that a space Cp(X) belongs to M if and only if it is a Lindelöf Σ-space. Under MA(ω1), if X is compact and (X×X)\Δ has a compact cover ordered by a Polish space then X is metrizable; here Δ={(x,x):xX} is the diagonal of the space X. Besides, if X is a compact space of countable tightness and X2\Δ belongs to M then X is metrizable in ZFC.We also consider the class M? of spaces X which have a compact cover F ordered by a second countable space with the additional property that, for every compact set PX there exists FF with PF. It is a ZFC result that if X is a compact space and (X×X)\Δ belongs to M? then X is metrizable. We also establish that, under CH, if X is compact and Cp(X) belongs to M? then X is countable.  相似文献   

17.
We prove Sobolev-type p(⋅)→q(⋅)-theorems for the Riesz potential operator Iα in the weighted Lebesgue generalized spaces Lp(⋅)(Rn,ρ) with the variable exponent p(x) and a two-parametrical power weight fixed to an arbitrary finite point and to infinity, as well as similar theorems for a spherical analogue of the Riesz potential operator in the corresponding weighted spaces Lp(⋅)(Sn,ρ) on the unit sphere Sn in Rn+1.  相似文献   

18.
In this paper we define model solvmanifold pairs and their diagonal type selfmaps in the tradition of Heath and Keppelmann. We derive an explicit formula for computing the relative Nielsen number N(F;X,A) on these spaces and selfmaps F:(X,A)→(X,A). We find that model solvmanifold pairs often exhibit interesting Schirmer theory, meaning N(F;X,A)>max{N(F),N(F|A)}.  相似文献   

19.
Let X and Y be separable Banach spaces and T:XY be a bounded linear operator. We characterize the non-separability of T?(Y?) by means of fixing properties of the operator T.  相似文献   

20.
Let GO(n) be a compact group of isometries acting on n-dimensional Euclidean space Rn, and X a bounded domain in Rn which is transformed into itself under the action of G. Consider a symmetric, classical pseudodifferential operator A0 in L2(Rn) with G-invariant Weyl symbol, and assume that it is semi-bounded from below. We show that the spectrum of the Friedrichs extension A of the operator is discrete, and derive asymptotics for the number Nχ(λ) of eigenvalues of A less or equal λ and with eigenfunctions in the χ-isotypic component of L2(X) as λ→∞, giving also an estimate for the remainder term in case that G is a finite group. In particular, we show that the multiplicity of each unitary irreducible representation in L2(X) is asymptotically proportional to its dimension.  相似文献   

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