Die invarianten Teilräume der stetigen Endomorphismen von ω

Körber, Karl-Heinz

doi:10.1007/bf01376216

Cited by 12 publications

(5 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is easy to see that every linear subspace of ϕ is closed, which leads to the following observation. It features (many times) in the literature, see, for instance, [16]. We present the proof for the sake of completeness.…”

Section: Introductionmentioning

confidence: 64%

Orbital and strongly orbital spaces

Shkarin

2014

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

We say that a (countably dimensional) topological vector space X is orbital if there is T ∈ L(X) and a vector x ∈ X such that X is the linear span of the orbit {T n x : n = 0, 1, . . . }. We say that X is strongly orbital if, additionally, x can be chosen to be a hypercyclic vector for T . Of course, X can be orbital only if the algebraic dimension of X is finite or infinite countable. We characterize orbital and strongly orbital metrizable locally convex spaces. We also show that every countably dimensional metrizable locally convex space X does not have the invariant subset property. That is, there is T ∈ L(X) such that every non-zero x ∈ X is a hypercyclic vector for T . Finally, assuming the Continuum Hypothesis, we construct a complete strongly orbital locally convex space.As a byproduct of our constructions, we determine the number of isomorphism classes in the set of dense countably dimensional subspaces of any given separable infinite dimensional Fréchet space X. For instance, in X = ℓ 2 × ω, there are exactly 3 pairwise non-isomorphic (as topological vector spaces) dense countably dimensional subspaces. MSC: 47A16

show abstract

Section: Introductionmentioning

confidence: 64%

Orbital and strongly orbital spaces

Shkarin

2014

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

show abstract

“…The result of Körber [17] and Shields [24] concerning the space ω directly follows from Theorem 2.1. We can in fact remark that a Fréchet space without continuous norm pX, pp j qq satisfies the assumptions of Theorem 2.1 if and only if X is isomorphic to ω ' Y where Y is a Fréchet space with continuous norm.…”

Section: Fréchet Spaces Satisfying the (Hereditary) Invariant Subspac...mentioning

confidence: 81%

“…We try in this paper to better understand which non-normable Fréchet spaces satisfy the (Hereditary) Invariant Subspaces/Subsets Properties. To this end, we first generalize the results obtained by Körber [17] and Shields [24] by showing that if Y is a Fréchet space with a continuous norm then ω ' Y satisfies the Invariant Subspace Property. We then look at the approach of Goliński concerning the existence of operators without non-trivial invariant subspaces.…”

Section: Introductionmentioning

confidence: 88%

“…In the case of non-normable Fréchet spaces, an example of separable infinitedimensional non-normable Fréchet space satisfying the Invariant Subspace Property is known for a long time. Indeed, It has been shown in 1969 by Körber [17] and later by Shields [24] that the space ω satisfies this property, where ω is the space of all complex sequences endowed with the seminorms p j pxq " max nďj |x n |. It was even shown by Johnson and Shields [16] that every operator on ω (that is not a scalar mulitple of the identity) possesses a non-trivial hyperinvariant subspace.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Invariant subspaces for non-normable Fréchet spaces

Menet

2018

Advances in Mathematics

View full text Add to dashboard Cite

A Fréchet space X satisfies the Hereditary Invariant Subspace (resp. Subset) Property if for every closed infinite-dimensional subspace M in X, each continuous operator on M possesses a non-trivial invariant subspace (resp. subset). In this paper, we exhibit a family of non-normable separable infinite-dimensional Fréchet spaces satisfying the Hereditary Invariant Subspace Property and we show that many non-normable Fréchet spaces do not satisfy this property. We also state sufficient conditions for the existence of a continuous operator without non-trivial invariant subset and deduce among other examples that there exists a continuous operator without non-trivial invariant subset on the space of entire functions HpCq.2010 Mathematics Subject Classification. 47A15; 47A16.

show abstract

“…The following results are contained in [8,9]: 1) to is the algebraic dual of the space ~0 of all sequences of complex numbers that have only finitely many nonzero elements. I.e., ~p = {{~k}e~: ek*0, for at most finitely many k's}.…”

Section: Transitive Subalgebras Of ~(~)mentioning

confidence: 99%

Burnside's theorem for the Fr�chet space ?

Herrero¹

1974

Math. Ann.

View full text Add to dashboard Cite

Die invarianten Teilräume der stetigen Endomorphismen von ω

Cited by 12 publications

References 8 publications

Orbital and strongly orbital spaces

Orbital and strongly orbital spaces

Invariant subspaces for non-normable Fréchet spaces

Burnside's theorem for the Fr�chet space ?

Contact Info

Product

Resources

About