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1.
In 1993, Y. A. Abramovich, C. D. Aliprantis and O. Burkinshaw showed that every continuous operator with modulus on an lp-space (1 ≤ p < ∞) whose modulus commutes with a non-zero positive operator T on lp that is quasinilpotent at a non-zero positive vector x0 has a non-trivial invariant closed subspace. In this paper, it is proved that if
is a collection of continuous operators with moduli on lp that is finitely modulus-quasinilpotent at a non-zero positive vector x
0 then
and its right modulus sub-commutant
have a common non-trivial invariant closed subspace. In particular, all continuous operators with moduli on l
p
whose moduli commute with a non-zero positive operator I on l
p
that is quasinilpotent at a non-zero positive vector x
0 have a common non-trivial invariant closed subspace, so that all positive operators on l
p
which commute with a non-zero positive operator S on l
p
that is quasinilpotent at a non-zero positive vector x
0 have a common non-trivial invariant closed subspace.
This research was supported by the Natural Science Foundation of Hunan Province of P. R. China (04JJ6004), the Foundation
of Education Department of Hunan Province of P. R. China (04C002) and the Natural Science Foundation of P. R. China (10671147).
Received: 4 December 2005 Revised: 19 June 2006 相似文献
2.
Roman Drnovšek 《Integral Equations and Operator Theory》2001,39(3):253-266
Let
be a collection of bounded operators on a Banach spaceX of dimension at least two. We say that
is finitely quasinilpotent at a vectorx
0X whenever for any finite subset
of
the joint spectral radius of
atx
0 is equal 0. If such collection
contains a non-zero compact operator, then
and its commutant
have a common non-trivial invariant, subspace. If in addition,
is a collection of positive operators on a Banach lattice, then
has a common non-trivial closed ideal. This result and a recent remarkable theorem of Turovskii imply the following extension of the famous result of de Pagter to semigroups. Let
be a multiplicative semigroup of quasinilpotent compact positive operators on a Banach lattice of dimension at least two. Then
has a common non-trivial invariant closed ideal.This work was supported by the Research Ministry of Slovenia. 相似文献
3.
Mohammed Taghi Jahandideh 《Proceedings of the American Mathematical Society》1997,125(9):2661-2670
There are some known results that guarantee the existence of a nontrivial closed invariant ideal for a quasinilpotent positive operator on an -space with unit or a Banach lattice whose positive cone contains an extreme ray. Some recent results also guarantee the existence of such ideals for certain positive operators, e.g. a compact quasinilpotent positive operator, on an arbitrary Banach lattice. The main object of this article is to use these results in constructing a maximal closed ideal chain, each of whose members is invariant under a certain collection of operators that are related to compact positive operators, or to quasinilpotent positive operators.
4.
LetT L(X) be a continuous linear operator on a complex Banach spaceX. We show thatT possesses non-trivial closed invariant subspaces if its localizable spectrum loc(T) is thick in the sense of the Scott Brown theory. Since for quotients of decomposable operators the spectrum and the localizable spectrum coincide, it follows that each quasiaffine transformation of a Banach-space operator with Bishop's property () and thick spectrum has a non-trivial invariant subspace. In particular it follows that invariant-subspace results previously known for restrictions and quotients of decomposable operators are preserved under quasisimilarity. 相似文献
5.
Coenraad C.A. Labuschagne 《Positivity》2006,10(2):391-407
Let E and F be Banach lattices and let S, T: E→ F be positive operators such that 0≤ S≤ T. It is shown that if T is a Radon–Nikodym operator, F has order continuous norm and E and F both have (Schaefer's) property (P), then S is a Radon–Nikodym operator; also, if T is an Asplund operator, E' has order continuous norm and E has property (P), then S is an Asplund operator. 相似文献
6.
A.K. Kitover 《Indagationes Mathematicae》2007,18(1):39-60
There are, by now, many results which guarantee that positive operators on Banach lattices have non-trivial closed invariant sublattices. In particular, this is true for every positive compact operator. Apart from some results of a general nature, in this paper we present several examples of positive operators on Banach lattices which do not have non-trivial closed invariant sublattices. These examples include both AM-spaces and Banach lattices with an order continuous norm and which are and are not atomic. In all these cases we can ensure that the operators do possess non-trivial closed invariant subspaces. 相似文献
7.
Ruey-Jen Jang 《Integral Equations and Operator Theory》2001,39(3):292-304
LetE be a Banach lattice with order continuous norm and {T(t)}
t0 be an eventually compactc
0-semigroup of positive operators onE with generator A. We investigate the structure of the geometric eigenspace of the generator belonging to the spectral bound when the semigroup is ideal reducible. It is shown that a basis of the eigenspace can be chosen to consist of element ofE with certain positivity structure. This is achieved by a decomposition of the underlying Banach latticeE into a direct sum of closed ideals which can be viewed as a generalization of the Frobenius normal form for nonnegative reducible matrices. 相似文献
8.
The size of the perturbation class {SL(E)S has closed range}+I(E) is studied, whereE is a Banach space andI(E) stands for various classical operator ideals. For instance, it is shown for the ideal consisting of the inessential operators that the resulting perturbation class does not exhaust the class of bounded linear operators under natural structural conditions onE. It is known from a recent result of Gowers and Maurey that some conditions are needed.Partially supported by the Academy of Finland 相似文献
9.
We obtain some necessary and some sufficient conditions on Banach lattices E and F for the following conditions to hold: (i) if T: E → F is a b-AM-compact operator, then T′: F′ → E′ is also b-AM-compact operator and (ii) if T′: F′ → E′ is b-AM-compact operator, then T: E → F is also b-AM-compact operator. 相似文献
10.
In this paper we find invariant subspaces of certain positive quasinilpotent operators on Krein spaces and, more generally,
on ordered Banach spaces with closed generating cones. In the later case, we use the method of minimal vectors. We present
applications to Sobolev spaces, spaces of differentiable functions, and C*-algebras.
相似文献
11.
Victor I. Lomonosov Heydar Radjavi Vladimir G. Troitsky 《Integral Equations and Operator Theory》2008,60(3):405-418
An algebra of operators on a Banach space X is said to be transitive if X has no nontrivial closed subspaces invariant under every member of the algebra. In this paper we investigate a number of
conditions which guarantee that a transitive algebra of operators is “large” in various senses. Among these are the conditions
of algebras being localizing or sesquitransitive. An algebra is localizing if there exists a closed ball B ∌ 0 such that for every sequence (x
n
) in B there exists a subsequence and a bounded sequence (A
k
) in the algebra such that converges to a non-zero vector. An algebra is sesquitransitive if for every non-zero z ∈ X there exists C > 0 such that for every x linearly independent of z, for every non-zero y ∈ X, and every there exists A in the algebra such that and ||Az|| ≤ C||z||. We give an algebraic version of this definition as well, and extend Jacobson’s density theorem to algebraically sesquitransitive
rings.
The second and the third authors were supported by NSERC. 相似文献
12.
《Quaestiones Mathematicae》2013,36(6):817-827
AbstractWe introduce and study the class of weak almost limited operators. We establish a characterization of pairs of Banach lattices E, F for which every positive weak almost limited operator T : E→F is almost limited (resp. almost Dunford- Pettis). As consequences, we will give some interesting results. 相似文献
13.
LetT be a positive linear operator on the Banach latticeE and let (S
n
) be a sequence of bounded linear operators onE which converge strongly toT. Our main results are concerned with the question under which additional assumptions onS
n
andT the peripheral spectra (S
n
) ofS
n
converge to the peripheral spectrum (T) ofT. We are able to treat even the more general case of discretely convergent sequences of operators. 相似文献
14.
During the last few years the authors have studied extensively the invariant subspace problem of positive operators; see [6] for a survey of this investigation. In [4] the authors introduced the class of compact-friendly operators and proved for them a general theorem on the existence of invariant subspaces. It was then asked if every positive operator is compact-friendly. In this note, we present an example of a positive operator which is not compact-friendly but which, nevertheless, has a non-trivial closed invariant subspace.In the process of presenting this example, we also characterize the multiplication operators that commute with non-zero finite-rank operators. We show, among other things, that a multiplication operator M
commutes with a non-zero finite-rank operator if and only the multiplier function is constant on some non-empty open set. 相似文献
15.
Let be a multiplicative semigroup of positive operators on a Banach lattice E such that every is ideal-triangularizable, i.e., there is a maximal chain of closed subspaces of E that consists of closed ideals invariant under S. We consider the question under which conditions the whole semigroup is simultaneously ideal-triangularizable. In particular, we extend a recent result of G. MacDonald and H. Radjavi. We also
introduce a class of positive operators that contains all positive abstract integral operators when E is Dedekind complete.
相似文献
16.
Peter Meyer-Nieberg 《Acta Appl Math》1992,27(1-2):91-100
In this paper we will discuss the local spectral behaviour of a closed, densely defined, linear operator on a Banach space. In particular, we are interested in closed, positive, linear operators, defined on an order dense ideal of a Banach lattice. Moreover, for positive, bounded, linear operators we will treat interpolation properties by means of duality.Dedicated to G. Maltese on the occasion of his 60th birthday 相似文献
17.
It is known that if a rearrangement invariant function space E on [0,1] has an unconditional basis then each linear continuous operator on E is a sum of two narrow operators. On the other hand, the sum of two narrow operators in L1 is narrow. To find a general approach to these results, we extend the notion of a narrow operator to the case when the domain
space is a vector lattice. Our main result asserts that the set Nr(E, F) of all narrow regular operators is a band in the vector lattice Lr(E, F) of all regular operators from a non-atomic order continuous Banach lattice E to an order continuous Banach lattice F. The band generated by the disjointness preserving operators is the orthogonal complement to Nr(E, F) in Lr(E, F). As a consequence we obtain the following generalization of the Kalton-Rosenthal theorem: every regular operator T : E → F from a non-atomic Banach lattice E to an order continuous Banach lattice F has a unique representation as T = TD + TN where TD is a sum of an order absolutely summable family of disjointness preserving operators and TN is narrow.
Supported by Ukr. Derzh. Tema N 0103Y001103. 相似文献
18.
Existence of Positive Solutions for Operator Equations and Applications to Semipositone Problems 总被引:2,自引:0,他引:2
In this paper we study the existence of positive solutions of the following operator equation in a Banach space E:
where G(x, λ) = λKFx+e0, K: E↦ E is a linear completely continuous operator, F: P↦ E is a nonlinear continuous , bounded operator, e0∈ E, λ is
a parameter and P is a cone of Banach space E. Since F is not assumed to be positive and e0 may be a negative element, the operator equation is a so-called semipositone problem. We prove that under certain super-linear
conditions on the operator F the operator equation has at least one positive solution for λ > 0 sufficiently small, and that under certain sub-linear
conditions on the operator F the operator equation has at least one positive solution for λ > 0 sufficiently large. In addition, we briefly outline an application of our results which simplify previous theorems in
the literature. 相似文献
19.
Let E and F be vector lattices and
the ordered space of all regular operators, which turns out to be a (Dedekind complete) vector lattice if F is Dedekind complete. We show that every lattice isomorphism from E onto F is a finite element in
, and that if E is an AL-space and F is a Dedekind complete AM-space with an order unit, then each regular operator is a finite element in
. We also investigate the finiteness of finite rank operators in Banach lattices. In particular, we give necessary and sufficient
conditions for rank one operators to be finite elements in the vector lattice
.
A half year stay at the Technical University of Dresden was supported by China Scholarship Council. 相似文献