共查询到20条相似文献,搜索用时 93 毫秒
1.
Alexey I. Popov 《Integral Equations and Operator Theory》2010,67(2):247-256
Given a Banach space X and a bounded linear operator T on X, a subspace Y of X is almost invariant under T if TY í Y+F{TY\subseteq Y+F} for some finite-dimensional “error” F. In this paper, we study subspaces that are almost invariant under every operator in an algebra
\mathfrak A{\mathfrak A} of operators acting on X. We show that if
\mathfrak A{\mathfrak A} is norm closed then the dimensions of “errors” corresponding to operators in
\mathfrak A{\mathfrak A} must be uniformly bounded. Also, if
\mathfrak A{\mathfrak A} is generated by a finite number of commuting operators and has an almost invariant half-space (that is, a subspace with both
infinite dimension and infinite codimension) then
\mathfrak A{\mathfrak A} has an invariant half-space. 相似文献
2.
Julio Flores 《Journal of Mathematical Analysis and Applications》2008,343(2):743-751
It is shown that every positive strictly singular operator T on a Banach lattice satisfying certain conditions is AM-compact and has invariant subspaces. Moreover, every positive operator commuting with T has an invariant subspace. It is also proved that on such spaces the product of a disjointly strictly singular and a regular AM-compact operator is strictly singular. Finally, we prove that on these spaces the known invariant subspace results for compact-friendly operators can be extended to strictly singular-friendly operators. 相似文献
3.
We say that a Banach space X satisfies the “descent spectrum equality” (in short, DSE) whenever, for every bounded linear operator T on X, the descent spectrum of T as an operator coincides with the descent spectrum of T as an element of the algebra of all bounded linear operators on X. We prove that the DSE is fulfilled by ℓ1, all Hilbert spaces, and all Banach spaces which are not isomorphic to any of their proper quotients (so, in particular,
by the hereditarily indecomposable Banach spaces [8]), but not by ℓ
p
, for 1 < p ≤ ∞ with p ≠ 2. Actually, a Banach space is not isomorphic to any of its proper quotients if and only if it is not isomorphic to any
of its proper complemented subspaces and satisfies the DSE. 相似文献
4.
Lower bounds are obtained for thegl constants and hence also for the unconditional basis constants of subspaces of finite dimensional Banach spaces. Sharp results
are obtained for subspaces ofl
∞
n
, while in the general case thegl constants of “random large” subspaces are related to the distance of “random large” subspaces to Euclidean spaces. In addition,
a new isometric characterization ofl
∞
n
is given, some new information is obtained concerningp-absolutely summing operators, and it is proved that every Banach space of dimensionn contains a subspace whose projection constant is of ordern
1/2.
The research for this paper was begun while both authors were guests of the Mittag-Leffler Institute.
Supported in part by NSF-MCS 79-03042. 相似文献
5.
A. K. Kitover 《Journal of Mathematical Sciences》1981,16(3):1184-1186
One considers “weighted translation” operators in ideal Banach spaces. It is proved that if the translation is aperiodic (the
set of periodic points has measure zero), then the spectrum of such an operator is rotationinvariant. This result can be extended
(under certain additional restrictions) to “weighted translation” operators acting in regular subspaces of ideal spaces, in
particular, to operators in Hardy spaces.
In this note we prove the rotation-invariance of the spectrum of aperiodic operators of “weighted translation” in ideal spaces
and uniform B-algebras.
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR,
Vol. 65, pp. 196–198, 1976. 相似文献
6.
O. V. Lopushans'kii 《Ukrainian Mathematical Journal》1992,44(4):443-453
A general method of constructing functions of unbounded operators acting in Banach spaces is set forth. It is based on equipping the initial space with invariant subspaces of ultradifferential vectors of a specified operator and describing the spectral properties of the initial space in the topological algebras which thereby arise.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 4, pp. 502–513, April, 1992. 相似文献
7.
Given an operator T : X → Y between Banach spaces, and a Banach lattice E consisting of measurable functions, we consider the point-wise extension of the operator to the vector-valued Banach lattices
T
E
: E(X) → E(Y) given by T
E
(f)(ω) = T(f(ω)). It is proved that for any Banach lattice E which does not contain c
0, the operator T is an isomorphism on a subspace isomorphic to c
0 if and only if so is T
E
. An analogous result for invertible operators on subspaces isomorphic to ℓ
1 is also given. 相似文献
8.
George Androulakis Alexey I. Popov Adi Tcaciuc Vladimir G. Troitsky 《Integral Equations and Operator Theory》2009,65(4):473-484
We introduce and study the following modified version of the Invariant Subspace Problem: whether every operator T on an infinite-dimensional Banach space has an almost invariant half-space, that is, a subspace Y of infinite dimension and infinite codimension such that Y is of finite codimension in T(Y). We solve this problem in the affirmative for a large class of operators which includes quasinilpotent weighted shift operators
on ℓp (1 ≤ p < ∞) or c0. 相似文献
9.
Special classes of intertwining transformations between Hilbert spaces are introduced and investigated, whose purposes are to provide partial answers to some classical questions on the existence of nontrivial invariant subspaces for operators acting on separable Hilbert spaces. The main result ensures that if an operator is \({{\mathcal D}}\)-intertwined to a normal operator, then it has a nontrivial invariant subspace. 相似文献
10.
Andrzej Kryczka 《Integral Equations and Operator Theory》2008,61(4):559-572
We introduce the arithmetic separation of a sequence—a geometric characteristic for bounded sequences in a Banach space which
describes the Banach-Saks property. We define an operator seminorm vanishing for operators with the Banach-Saks property.
We prove quantitative stability of the seminorm for a class of operators acting between l
p
-sums of Banach spaces. We show logarithmically convex-type estimates of the seminorm for operators interpolated by the real
method of Lions and Peetre.
相似文献
11.
This paper firstly discusses the existence of strongly irreducible operators on Banach spaces. It shows that there exist strongly
irreducible operators on Banach spaces with w*-separable dual. It also gives some properties of strongly irreducible operators on Banach spaces. In particular, if T is a strongly irreducible operator on an infinite-dimensional Banach space, then T is not of finite rank and T is not an algebraic operator. On Banach spaces with subsymmetric bases, including infinite-dimensional separable Hilbert
spaces, it shows that quasisimilarity does not preserve strong irreducibility. In addition, we show that the strong irreducibility
of an operator does not imply the strong irreducibility of its conjugate operator, which is not the same as the property in
Hilbert spaces. 相似文献
12.
Let T be a bounded linear operator on Hilbert space H, M an invariant subspace of T. If there exists another invariant subspace N of T such that H = M + N and M ∩ N = 0, then M is said to be a completely reduced subspace of T. If T has a nontrivial completely reduced subspace, then T is said to be completely reducible; otherwise T is said to be completely irreducible. In the present paper we briefly sum up works on completely irreducible operators that
have been done by the Functional Analysis Seminar of Jilin University in the past ten years and more.
The paper contains four sections. In section 1 the background of completely irreducible operators is given in detail. Section
2 shows which operator in some well-known classes of operators, for example, weighted shifts, Toeplitz operators, etc., is
completely irreducible. In section 3 it is proved that every bounded linear operator on the Hilbert space can be approximated
by the finite direct sum of completely irreducible operators. It is clear that a completely irreducible operator is a rather
suitable analogue of Jordan blocks in L(H), the set of all bounded linear operators on Hilbert space H. In section 4 several questions concerning completely irreducible operators are discussed and it is shown that some properties
of completely irreducible operators are different from properties of unicellular operators.
__________
Translated from Acta Sci. Nat. Univ. Jilin, 1992, (4): 20–29 相似文献
13.
The invariant subspace problem for a class of Banach spaces, 2: Hypercyclic operators 总被引:2,自引:0,他引:2
C. J. Read 《Israel Journal of Mathematics》1988,63(1):1-40
We continue here the line of investigation begun in [7], where we showed that on every Banach spaceX=l
1⊗W (whereW is separable) there is an operatorT with no nontrivial invariant subspaces. Here, we work on the same class of Banach spaces, and produce operators which not
only have no invariant subspaces, but are also hypercyclic. This means that for every nonzero vectorx inX, the translatesT
r x (r=1, 2, 3,...) are dense inX. This is an interesting result even if stated in a form which disregards the linearity ofT: it tells us that there is a continuous map ofX{0\{ into itself such that the orbit {T
rx :r≧0{ of anyx teX \{0\{ is dense inX \{0\{. The methods used to construct the new operatorT are similar to those in [7], but we need to have somewhat greater complexity in order to obtain a hypercyclic operator. 相似文献
14.
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
15.
A new method of defining hereditarily indecomposable Banach spaces is presented. This method provides a unified approach for constructing reflexive HI spaces and also HI spaces with no reflexive subspace. All the spaces presented here satisfy the property that the composition of any two strictly singular operators is a compact one. This yields the first known example of a Banach space with no reflexive subspace such that every operator has a non-trivial closed invariant subspace. 相似文献
17.
W. E. Longstaff 《Integral Equations and Operator Theory》2003,45(3):343-350
In complex, separable, infinite-dimensional Hilbert space there exist
5 proper dense operator ranges with the property that every operator leaving
each of them invariant is a scalar multiple of the identity. The algebra of
operators leaving a pair of proper dense operator ranges invariant can have
an infinite nest of invariant subspaces. A slight extension of Foiaş' Theorem
shows that it can also have a non-trivial reducing subspace.
Submitted: July 13, 2001? Revised: December 6, 2001. 相似文献
18.
《Comptes Rendus de l'Academie des Sciences Series IIA Earth and Planetary Science》2001,332(2):151-156
We show that the consideration of Gâteaux smooth functions on Banach spaces which admit an equivalent Gâteaux smooth norm allows us to show that certain linear operators have nontrivial closed invariant subspaces. It is in particular the case of all operators on a real Banach space which admit a moment sequence. 相似文献
19.
D. S. Anisimov 《Journal of Mathematical Sciences》2006,139(2):6363-6368
A version of Grothendieck’s inequality says that any bounded linear operator acting from a Banach lattice X to a Banach lattice
Y acts from X(ℓ2) to Y (ℓ2) as well. A similar statement is proved for Hardy-type subspaces in lattices of measurable functions. Namely, let X be a
Banach lattice of measurable functions on the circle, and let an operator T act from the corresponding subspace of analytic
functions XA to a Banach lattice Y or, if Y is also a lattice of measurable functions on the circle, to the quotient space Y/YA. Under certain mild conditions on the lattices involved, it is proved that T induces an operator acting from XA(ℓ2) to Y (ℓ2) or to Y/YA(ℓ2), respectively. Bibliography: 7 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 327, 2005, pp. 5–16. 相似文献
20.
Jin Chuan HOU Xiu Ling ZHANG 《数学学报(英文版)》2006,22(1):179-186
Let X be an infinite-dimensional complex Banach space and denote by B(X) the algebra of all bounded linear operators acting on X. It is shown that a surjective additive map Φ from B(X) onto itself preserves similarity in both directions if and only if there exist a scalar c, a bounded invertible linear or conjugate linear operator A and a similarity invariant additive functional ψ on B(X) such that either Φ(T) = cATA^-1 + ψ(T)I for all T, or Φ(T) = cAT*A^-1 + ψ(T)I for all T. In the case where X has infinite multiplicity, in particular, when X is an infinite-dimensional Hilbert space, the above similarity invariant additive functional ψ is always zero. 相似文献