Invariant subspace problem for classical spaces of functions

Goliński, Michał

doi:10.1016/j.jfa.2011.11.003

Cited by 11 publications

(14 citation statements)

References 9 publications

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“…We remark that Corollary 3.7 does not rely on the structure of the norm p j but only on values of norms p j pe n q. In particular, unlike results in [10], this corollary can be applied to Köthe spaces Λ p pAq for every p ě 1 and not only for p " 1. In particular, we can apply Corollary 3.7 to power series spaces without assuming the nuclearity.…”

Section: Fréchet Spaces Not Satisfying the (Hereditary) Invariant Submentioning

confidence: 93%

“…On the other hand, if X is a Fréchet space without continuous norm satisfying the conditions of Theorem 2.1 then ker p j0 is isomorphic to ω (see [19,Proposition 26.16]) and thus X is isomorphic to ω ' Y where Y is a Fréchet space with continuous norm. Thanks to Atzmon [3,4] and Goliński [10,11], we know several examples of non-normable Fréchet spaces which do not satisfy the Invariant Subset Property or the Invariant Subspace Property. In the next section, we investigate sufficient conditions for having no non-trivial invariant subset by trying to take the best advantage of the non-normability of non-normable Fréchet spaces.…”

Section: Fréchet Spaces Satisfying the (Hereditary) Invariant Subspacmentioning

confidence: 99%

“…For instance, if we consider X " HpCq endowed with the norms p j pf q " sup |z|ďj |f pzq| and the derivative operator D, we have p j pDf q ď p j`1 pf q for every f P HpCq and p j pDz n q " nj n´1 " n j p j pz n q for every n ě 1. This particularity of non-normable Fréchet spaces is not used in [10] since the considered operators satisfy p j pT xq ď C j p j pxq for every j ě 1. Due to this fact, Goliński can only consider Fréchet spaces with ℓ 1 -norms.…”

Section: Fréchet Spaces Not Satisfying the (Hereditary) Invariant Submentioning

confidence: 99%

“…In [10], it is also shown that HpCq does not satisfy the Invariant Subspace Property. Thanks to Corollary 3.7, we improve this result by showing that HpCq does not satisfy the Invariant Subset Property.…”

Section: Fréchet Spaces Not Satisfying the (Hereditary) Invariant Submentioning

confidence: 99%

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Invariant subspaces for non-normable Fréchet spaces

Menet

2018

Advances in Mathematics

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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.

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Section: Fréchet Spaces Not Satisfying the (Hereditary) Invariant Submentioning

confidence: 93%

Section: Fréchet Spaces Satisfying the (Hereditary) Invariant Subspacmentioning

confidence: 99%

Section: Fréchet Spaces Not Satisfying the (Hereditary) Invariant Submentioning

confidence: 99%

Section: Fréchet Spaces Not Satisfying the (Hereditary) Invariant Submentioning

confidence: 99%

See 2 more Smart Citations

Invariant subspaces for non-normable Fréchet spaces

Menet

2018

Advances in Mathematics

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“…If fact, C. Read [243] solved the invariant subset problem (analogous to the invariant subspace problem, connected with cyclic operators) for Banach spaces (it remains unsolved for Hilbert spaces) by exhibiting an operator T on the sequence space 1 for which any nonzero vector is hypercyclic. (Incidentally, in the recent paper [168] Goliński has given examples of operators S without nontrivial proper invariant subspaces on classical non-Banach spaces X; hence the set of S-cyclic vectors is X \ {0}. )…”

Section: Hypercyclity and Dense-lineability An Extreme Case Of Lineamentioning

confidence: 99%

Linear subsets of nonlinear sets in topological vector spaces

Bernal–González

Pellegrino

Seoane‐Sepúlveda

2013

Bull. Amer. Math. Soc.

197

149

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For the last decade there has been a generalized trend in mathematics on the search for large algebraic structures (linear spaces, closed subspaces, or infinitely generated algebras) composed of mathematical objects enjoying certain special properties. This trend has caught the eye of many researchers and has also had a remarkable influence in real and complex analysis, operator theory, summability theory, polynomials in Banach spaces, hypercyclicity and chaos, and general functional analysis. This expository paper is devoted to providing an account on the advances and on the state of the art of this trend, nowadays known as lineability and spaceability. 109 5. Some remarks and conclusions. General techniques 113 About the authors 117 Acknowledgments 118 References 118Theorem 1.2 (Gurariy, 1966 [178]). The set of continuous nowhere differentiable functions on [0, 1] is lineable.Let us also recall that, although the set of everywhere differentiable functions in R is, in itself, an infinite dimensional vector space, in 1966 Gurariy obtained the following analogue of Theorem 1.1.License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use LINEAR SUBSETS OF NONLINEAR SETS IN TOPOLOGICAL VECTOR SPACES 73 Theorem 1.3 (Gurariy, 1966 [178]). The set of everywhere differentiable functions on [0, 1] is not spaceable in C[0, 1]. Also, there exist closed infinite dimensional subspaces of C[0, 1] all of whose members are differentiable on (0, 1).Somehow, what we are seeing is that what one could expect to be an isolated phenomenon can actually even have a nice algebraic structure (in the form of infinite dimensional subspaces). Unfortunately, and as we mentioned above, the Baire category theorem cannot be employed in the search for large subspaces such as the ones mentioned in the previous results. Let us now provide a more formal and complete definition for the above concepts and some other ones. Definition 1.4 (Lineability and spaceability [22,181,253]). Let X be a topological vector space and M a subset of X. Let μ be a cardinal number.(1) M is said to be μ-lineable (μ-spaceable) if M ∪ {0} contains a vector space (resp. a closed vector space) of dimension μ. At times, we shall refer to the set M as simply lineable or spaceable if the existing subspace is infinite dimensional.(2) We also let λ(M ) be the maximum cardinality (if it exists) of such a vector space. 1 (3) When the above linear space can be chosen to be dense in X, we shall say that M is μ-dense-lineable.Moreover, Bernal introduced in [69] the notion of maximal-lineable (and those of maximal-dense-lineable and maximal-spaceable), meaning that, when keeping the above notation, the dimension of the existing linear space equals dim(X). Besides asking for linear spaces, one could also study other structures, such as algebrability and some related ones, which were presented in [25,27,253]. Definition 1.5. Given a Banach algebra A, a subset B ⊂ A, and two cardinal numbers α and β, we say that:

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Operator on the space of rapidly decreasing functions with all non-zero vectors hypercyclic

Goliński

2013

Advances in Mathematics

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Invariant subspace problem for classical spaces of functions

Cited by 11 publications

References 9 publications

Invariant subspaces for non-normable Fréchet spaces

Invariant subspaces for non-normable Fréchet spaces

Linear subsets of nonlinear sets in topological vector spaces

Operator on the space of rapidly decreasing functions with all non-zero vectors hypercyclic

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