An Operator Without Invariant Subspaces on a Nuclear Frechet Space

Atzmon, Aharon

doi:10.2307/2007039

Cited by 30 publications

(19 citation statements)

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“…Enflo [6] and Read [17] constructed separable Banach spaces which support an operator without closed nontrivial invariant subspaces. An operator without non-trivial invariant subspaces on a nuclear Fre chet space was constructed by Atzmon [2]. For more information we refer to the introduction of [9].…”

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confidence: 99%

Hypercyclic Operators on Non-normable Fréchet Spaces

Bonet

Peris

1998

Journal of Functional Analysis

108

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Every infinite dimensional separable non-normable Fre chet space admits a continuous hypercyclic operator. A large class of separable countable inductive limits of Banach spaces with the same property is given, but an example of a separable complete inductive limit of Banach spaces which admits no hypercyclic operator is provided. It is also proved that no compact operator on a locally convex space is hypercyclic.

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mentioning

confidence: 99%

Hypercyclic Operators on Non-normable Fréchet Spaces

Bonet

Peris

1998

Journal of Functional Analysis

108

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“…Read was also even able to produce an example of operator with no nontrivial invariant subset [54]. Closely related to our problem, we should also mention that Atzmon [7] was the first to construct an example of operator on a nuclear Fréchet space with no ntis.…”

Section: Ifmentioning

confidence: 77%

An overview of some recent developments on the Invariant Subspace Problem

Chalendar

Partington

2012

Concrete Operators

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This paper presents an account of some recent approaches to the Invariant Subspace Problem. It contains a brief historical account of the problem, and some more detailed discussions of specific topics, namely, universal operators, the Bishop operators, and Read's Banach space counter-example involving a finitely strictly singular operator. History of the problemThere is an important problem in operator theory called the invariant subspace problem. This problem is open for more than half a century, and there are many significant contributions with a huge variety of techniques, making this challenging problem so interesting; however the solution seems to be nowhere in sight. In this paper we first review the history of the problem, and then present an account of some recent developments with which we have been involved. Other recent approaches are discussed in [21]. The invariant subspace problem is the following: let be a complex Banach space of dimension at least 2 and let T ∈ ( ), i.e., T : → , linear and bounded. Is there any closed subspace ⊂ such that T ( ) ⊂ (i.e., is invariant for T ) and = {0} = (i.e., is not trivial)? In the sequel a "nontrivial invariant subspace" may be abbreviated as "ntis".• Here is a list of preliminary remarks:Assume that is of finite dimension ≥ 2, so that is isomorphic to C . Then T ∈ ( ) is a × matrix with complex entries, and thus T has eigenvectors. Each eigenvector generates a (nontrivial) invariant subspace of dimension 1. The Jordan form of T is of great help in describing the lattice of the invariant subspaces of T . Ifis not separable, then T ∈ ( ) has a nontrivial invariant subspace since for all ∈ , = 0, the closure of { (T ) : ∈ C[ ]} is invariant for T and nontrivial since it is a separable subspace by construction, containing a nonzero vector.Note also that for real Banach spaces, there exist operators with no nontrivial invariant subspace: take = R 2 and T a rotation of angle θ ∈ (0 2π) \

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“…A commutative semitopological algebra is topologically simple if it has no closed proper non-zero ideals. Examples of commutative unital complete non-metrizable locally convex topologically simple Hausdorff algebras have been given in [13] and in [31]. It is shown in [31], Proposition 1, that every commutative unital topological algebra is topologically simple if and only if G t (A) = A \ {θ A }.…”

Section: Examples Of T Q-algebrasmentioning

confidence: 99%

“…It is shown in [31], Proposition 1, that every commutative unital topological algebra is topologically simple if and only if G t (A) = A \ {θ A }. Hence, topological algebras in [13] and in [31] described above are commutative T Q-algebras.…”

Section: Examples Of T Q-algebrasmentioning

confidence: 99%

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Properties ofTQ-algebras

Abel¹,

Żelazko²

2011

Proc. Estonian Acad. Sci.

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Several properties of unital left (right) T Q-algebras are described. The conditions when a unital semitopological algebra is a left (right) T Q-algebra are given. It is shown that the space M(A) (of nontrivial continuous multiplicative linear functionals on A) in the Gelfand topology is a compact Hausdorff space for every unital T Q-algebra with a nonempty set M(A) and a commutative complete metrizable unital algebra is a T Q-algebra if and only if all maximal topological ideals of A are closed. Examples of T Q-algebras are given. Open problems are presented. Here and later on every U denotes the closure of U in A.2 That is, the net (a λ ) λ ∈Λ is the same in (a λ a) λ ∈Λ and (aa λ ) λ ∈Λ . 3 It is known (see [2], Corollary 2) that every complete unital locally m-pseudoconvex algebra is an invertive algebra, but every commutative complete metrizable unital algebra with a discontinuous inverse is not (see [32], Proposition 4).

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An Operator Without Invariant Subspaces on a Nuclear Frechet Space

Cited by 30 publications

References 11 publications

Hypercyclic Operators on Non-normable Fréchet Spaces

Hypercyclic Operators on Non-normable Fréchet Spaces

An overview of some recent developments on the Invariant Subspace Problem

Properties ofTQ-algebras

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