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1.
On weak positive supercyclicity 总被引:1,自引:0,他引:1
A bounded linear operator T on a separable complex Banach space X is called weakly supercyclic if there exists a vector x ∈ X such that the projective orbit {λT
n
x: n ∈ ℕ λ ∈ ℂ} is weakly dense in X. Among other results, it is proved that an operator T such that σ
p
(T
*) = 0, is weakly supercyclic if and only if T is positive weakly supercyclic, that is, for every supercyclic vector x ∈ X, only considering the positive projective orbit: {rT
n
x: n ∈ ℂ, r ∈ ℝ+} we obtain a weakly dense subset in X. As a consequence it is established the existence of non-weakly supercyclic vectors (non-trivial) for positive operators
defined on an infinite dimensional separable complex Banach space. The paper is closed with concluding remarks and further
directions.
Partially supported by MEC MTM2006-09060 and MTM2006-15546, Junta de Andalucía FQM-257 and P06-FQM-02225.
Partially supported by Junta de Andalucía FQM-257, and P06-FQM-02225 相似文献
2.
We give a direct, self-contained, and iterative proof that for any convex, Lipschitz andw
*-lower semicontinuous function ϕ defined on aw
*-compact convex setC in a dual Banach spaceX
* and for any ε>0 there is anx∈X, with ‖x‖≤ε, such that ϕ+x attains its supremum at an extreme point ofC. This result is implicitly contained in the work of Lindenstrauss [9] and the work of Ghoussoub and Maurey on strongw
*−H
σ sets [8]. In addition, we discuss the applications of this result to the geometry of convex sets.
Research supported in part by the NSERC of Canada under grant OGP41983 for the first author and grant OGP7926 for the second
author. 相似文献
3.
In this paper we apply Bishop-Phelps property to show that if X is a Banach space and G X is the maximal subspace so that G⊥ = {x* ∈ X*|x*(y) = 0; y∈ G} is an L-summand in X*, then L1(Ω,G) is contained in a maximal proximinal subspace of L1(Ω,X). 相似文献
4.
Vladimír Müller 《Integral Equations and Operator Theory》2001,41(2):230-253
LetT be an operator on a Banach spaceX. We give a survey of results concerning orbits {T
n
x:n=0,1,...} and weak orbits {T
n
x,x
*:n=0,1,...} ofT wherexX andx
*X
*. Further we study the local capacity of operators and prove that there is a residual set of pointsxX with the property that the local capacity cap(T, x) is equal to the global capacity capT. This is an analogy to the corresponding result for the local spectral radius.The research was supported by the grant No. A1019801 of AV R. 相似文献
5.
Suppose that(T
t
)t>0 is aC
0 semi-group of contractions on a Banach spaceX, such that there exists a vectorx∈X, ‖x‖=1 verifyingJ
−1(Jx)={x}, whereJ is the duality mapping fromX toP(X
*). If |<T
t
x,f>|→1, whent→+∞ for somef∈X
*, ‖f‖≤1 thenx is an eigenvector of the generatorA, associated with a purcly imaginary eigenvalue. Because of Lin's example [L], the hypothesis onx∈X is the best possible.
If the hypothesisJ
−1(Jx)={x} is not verified, we can prove that ifJx is a singleton and ifJ
−1(Jx) is weakly compact, then if |<T
t
x, f>|→1, whent→+∞ for somef∈X
*, ‖f‖≤1, there existsy∈J
−1(Jx) such thaty is an eigenvector of the generatorA, associated with a purely imaginary eigenvalue. We give also a counter-example in the case whereX is one of the spaces ℓ1 orL
1. 相似文献
6.
We are going to discuss special cases of a conditional functional inequality
whereX is a real inner product space. In particular, we will give conditions which force the representationf(x)=c‖x‖2+a(x) for x ∈X, where c ∈ R anda:x→ℝ is an additive functional. 相似文献
7.
B. P. Duggal 《Rendiconti del Circolo Matematico di Palermo》2007,56(3):317-330
A Banach space operatorT ɛB(X) is polaroid,T ɛP, if the isolated points of the spectrum ofT are poles of the resolvent ofT. LetPS denote the class of operators inP which have have SVEP, the single-valued extension property. It is proved that ifT is polynomiallyPS andA ɛB(X) is an algebraic operator which commutes withT, thenf(T+A) satisfies Weyl’s theorem andf(T
*+A
*) satisfiesa-Weyl’s theorem for everyf which is holomorphic on a neighbourhood of σ(T+A). 相似文献
8.
W. M. Ruess 《Semigroup Forum》1995,51(1):335-341
For aC
0-contraction semigroup (S(t))
t≥0 of bounded linear operators on a complex Banach spaceX, J. A. Goldstein and B. Nagy [6] have shown that, givenx∈X, S(t)x=e
iλt
x, t≥0, for some λ∈ℝ, provided lim
t→∞
|<S(t)x,x
*
>|=|<x,x
*
>| for allx
*∈X*. We present (a) an extension to the case of nonlinear nonexpansive mapsS(t), t≥0, and (b) various generalizations in the linear context. 相似文献
9.
Let X and Y be Banach spaces. We say that a set
(the space of all weakly compact operators from X into Y) is weakly equicompact if, for every bounded sequence (xn) in X, there exists a subsequence (xk(n)) so that (Txk(n)) is weakly uniformly convergent for T ∈ M. We study some properties of weakly equicompact sets and, among other results, we prove: 1) if
is collectively weakly compact, then M* is weakly equicompact iff M** x**={T** x** : T ∈ M} is relatively compact in Y for every x** ∈X**; 2) weakly equicompact sets are precompact in
for the topology of uniform convergence on the weakly null sequences in X.
Received: 14 February 2005; revised: 1 June 2005 相似文献
10.
Let [A, a] be a normed operator ideal. We say that [A, a] is boundedly weak*-closed if the following property holds: for all Banach spaces X and Y, if T: X → Y** is an operator such that there exists a bounded net (T
i
)
i∈I
in A(X, Y) satisfying lim
i
〈y*, T
i
x
y*〉 for every x ∈ X and y* ∈ Y*, then T belongs to A(X, Y**). Our main result proves that, when [A, a] is a normed operator ideal with that property, A(X, Y) is complemented in its bidual if and only if there exists a continuous projection from Y** onto Y, regardless of the Banach space X. We also have proved that maximal normed operator ideals are boundedly weak*-closed but, in general, both concepts are different.
相似文献
11.
Pei-Kee Lin 《Semigroup Forum》1996,53(1):208-211
For any complex Banach spaceX, letJ denote the duality mapping ofX. For any unit vectorx inX and any (C
0) contraction semigroup (T
t
)
t>0 onX, Baillon and Guerre-Delabriere proved that ifX is a smooth reflexive Banach space and if there isx
*∈J(x) such that ÷〈(T(t)x, J(x)〈÷→1 ast→∞, then there is a unit vectory∈X which is an eigenvector of the generatorA of (T
t
)
t>0 associated with a purely imaginary eigenvalue. They asked whether this result is still true ifX is replaced byc
0. In this article, we show the answer is negative
Partial results of this paper were obtained when the author attended the International Conference of Convexity at the University
of Marne-La-Vallée. He would like to express his gratitude for the kind hospitality offered to him. He would also like to
thank Profs. Goldstein and Jamison for their valuable suggestions. 相似文献
12.
LetT be a nonexpansive mapping on a normed linear spaceX. We show that there exists a linear functional.f, ‖f‖=1, such that, for allx∈X, limn→x
f(T
n
x/n)=limn→x‖T
n
x/n
‖=α, where α≡inf
y∈c
‖Ty-y‖. This means, ifX is reflexive, that there is a faceF of the ball of radius α to whichT
n
x/n converges weakly for allx (infz∈f
g(T
n
x/n-z)→0, for every linear functionalg); ifX is strictly conves as well as reflexive, the convergence is to a point; and ifX satisfies the stronger condition that its dual has Fréchet differentiable norm then the convergence is strong. Furthermore,
we show that each of the foregoing conditions on X is satisfied if and only if the associated convergence property holds for
all nonexpansiveT.
Supported by National Science Foundation Grant MCS-79-066. 相似文献
13.
We introduce a geometrical property of norm one complemented subspaces ofC(K) spaces which is useful for computing lower bounds on the norms of projections onto subspaces ofC(K) spaces. Loosely speaking, in the dual of such a space ifx* is a w* limit of a net (x
a
*
) andx*=x*1+x*2 with ‖x*‖=‖x*1‖ + ‖x*2‖, then we measure how efficiently thex
a
*
's can be split into two nets converging tox*1 andx*2, respectively. As applications of this idea we prove that if for everyε>0,X is a norm (1+ε) complemented subspace of aC(K) space, then it is norm one complemented in someC(K) space, and we give a simpler proof that a slight modification of anl
1-predual constructed by Benyamini and Lindenstrauss is not complemented in anyC(K) space.
Research partially supported by a grant of the U.S.-Israel Binational Science Foundation.
Research of the first-named author is supported in part by NSF grant DMS-8602395.
Research of the second-named author was partially supported by the Fund for the Promotion of Research at the Technion, and
by the Technion VPR-New York Metropolitan Research Fund. 相似文献
14.
K. V. Storozhuk 《Siberian Mathematical Journal》2011,52(6):1104-1107
Let X be a Banach space and let T: X → X be a power bounded linear operator. Put X
0 = {x ∈ X ∣ T
n
x → 0}. Assume given a compact set K ⊂ X such that lim inf
n→∞
ρ{T
n
x, K} ≤ η < 1 for every x ∈ X, ∥x∥ ≤ 1. If $\eta < \tfrac{1}
{2}
$\eta < \tfrac{1}
{2}
, then codim X
0 < ∞. This is true in X reflexive for $\eta \in [\tfrac{1}
{2},1)
$\eta \in [\tfrac{1}
{2},1)
, but fails in the general case. 相似文献
15.
A Banach space operatorT ∈B(χ) is said to behereditarily normaloid, denotedT ∈ ℋN, if every part ofT is normaloid;T ∈ ℋN istotally hereditarily normaloid, denotedT ∈ ℑHN, if every invertible part ofT is also normaloid. Class ℑHN is large; it contains a number of the commonly considered classes of operators. The operatorT isalgebraically totally hereditarily normaloid, denotedT ∈a — ℑHN, both non-constant polynomialp such thatp(T) ∈ ℑHN. For operatorsT ∈a − ℑHN, bothT andT* satisfy Weyl’s theorem; if also either ind(T−μ)≥0 or ind(T−μ)≤0 for all complexμ such thatT−μ is Fredholm, thenf(T) andf(T*) satisfy Weyl’s theorem for all analytic functionsf ∈ ℋ(σ(T)). For operatorsT ∈a — ℑHN such thatT has SVEP,T* satisfiesa-Weyl’s theorem. 相似文献
16.
Haïkel Skhiri 《Acta Appl Math》2010,112(3):347-356
Let m(T) and q(T) be respectively the minimum and the surjectivity moduli of T∈ℬ(X), where ℬ(X) denotes the algebra of all bounded linear operators on a complex Banach space X. If there exists a semi-invertible but non-invertible operator in ℬ(X) then, given a surjective unital linear map φ: ℬ(X)⟶ℬ(X), we prove that m(T)=m(φ(T)) for all T∈ℬ(X), if and only if, q(T)=q(φ(T)) for all T∈ℬ(X), if and only if, there exists a bijective isometry U∈ℬ(X) such that φ(T)=UTU
−1 for all T∈ℬ(X). 相似文献
17.
We consider the general optimization problem (P) of selecting a continuous function x over a -compact Hausdorff space T to a metric space A, from a feasible region X of such functions, so as to minimize a functional c on X. We require that X consist of a closed equicontinuous family of functions lying in the product (over T) of compact subsets Y
t
of A. (An important special case is the optimal control problem of finding a continuous time control function x that minimizes its associated discounted cost c(x) over the infinite horizon.) Relative to the uniform-on-compacta topology on the function space C(T,A) of continuous functions from T to A, the feasible region X is compact. Thus optimal solutions x
* to (P) exist under the assumption that c is continuous. We wish to approximate such an x
* by optimal solutions to a net {P
i
}, iI, of approximating problems of the form minxX
i
c
i(x) for each iI, where (1) the net of sets {X
i
}
I
converges to X in the sense of Kuratowski and (2) the net {c
i
}
I
of functions converges to c uniformly on X. We show that for large i, any optimal solution x
*
i
to the approximating problem (P
i
) arbitrarily well approximates some optimal solution x
* to (P). It follows that if (P) is well-posed, i.e., limsupX
i
* is a singleton {x
*}, then any net {x
i
*}
I
of (P
i
)-optimal solutions converges in C(T,A) to x
*. For this case, we construct a finite algorithm with the following property: given any prespecified error and any compact subset Q of T, our algorithm computes an i in I and an associated x
i
* in X
i
* which is within of x
* on Q. We illustrate the theory and algorithm with a problem in continuous time production control over an infinite horizon. 相似文献
18.
We show that to each asymptotic contraction T with a bounded orbit in a complete metric space X, there corresponds a unique point x
* such that all the iterates of T converge to x
*, uniformly on any bounded subset of X. If, in addition, some power of T is continuous at x
*, then x
* is a fixed point of T.
Dedicated to Professor Felix E. Browder with admiration and respect 相似文献
19.
Dieter Bothe 《Israel Journal of Mathematics》1998,108(1):109-138
Given anm-accretive operatorA in a Banach spaceX and an upper semicontinuous multivalued mapF: [0,a]×X→2
X
, we consider the initial value problemu′∈−Au+F(t,u) on [0,a],u(0)=x
0. We concentrate on the case when the semigroup generated by—A is only equicontinuous and obtain existence of integral solutions if, in particular,X* is uniformly convex andF satisfies β(F(t,B))≤k(t)β(B) for all boundedB⊂X wherek∈L
1([0,a]) and β denotes the Hausdorff-measure of noncompactness. Moreover, we show that the set of all solutions is a compactR
δ-set in this situation. In general, the extra condition onX* is essential as we show by an example in whichX is not uniformly smooth and the set of all solutions is not compact, but it can be omited ifA is single-valued and continuous or—A generates aC
o-semigroup of bounded linear operators. In the simpler case when—A generates a compact semigroup, we give a short proof of existence of solutions, again ifX* is uniformly (or strictly) convex. In this situation we also provide a counter-example in ℝ4 in which no integral solution exists.
The author gratefully acknowledges financial support by DAAD within the scope of the French-German project PROCOPE. 相似文献
20.
Domingo Pestana José M. Rodríguez José M. Sigarreta María Villeta 《Central European Journal of Mathematics》2012,10(3):1141-1151
If X is a geodesic metric space and x
1; x
2; x
3 ∈ X, a geodesic triangle T = {x
1; x
2; x
3} is the union of the three geodesics [x
1
x
2], [x
2
x
3] and [x
3
x
1] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. We denote by δ(X) the sharp hyperbolicity constant of X, i.e., δ(X) = inf {δ ≥ 0: X is δ-hyperbolic}. We obtain information about the hyperbolicity constant of cubic graphs (graphs with all of their vertices of
degree 3), and prove that for any graph G with bounded degree there exists a cubic graph G* such that G is hyperbolic if and only if G* is hyperbolic. Moreover, we prove that for any cubic graph G with n vertices, we have δ(G) ≤ min {3n/16 + 1; n/4}. We characterize the cubic graphs G with δ(G) ≤ 1. Besides, we prove some inequalities involving the hyperbolicity constant and other parameters for cubic graphs. 相似文献