On typical properties of Hilbert space operators

Eisner, Tanja; Mátrai, Tamás

doi:10.1007/s11856-012-0128-7

Cited by 13 publications

(28 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This theorem is proved in [13] by showing that a typical T P pB 1 pHq, SOTq has the following properties: T˚is an isometry, dim kerpT q " 8, and }T n x} Ñ 0 as n Ñ 8 for every x P H. The result then follows from the Wold decomposition theorem. In particular, it follows immediately from (c) that an SOT -typical contraction on H has a wealth of invariant subspaces.…”

Section: Proof Ofmentioning

confidence: 94%

“…The study of typical properties of contractions was initiated, in a Hilbertian setting, by Eisner [12] and Eisner-Mátrai [13]. Given a Hilbert space H (complex, infinite-dimensional and separable), Eisner studied in [12] properties of typical operators T P B 1 pHq for the Weak Operator Topology and proved that a typical T P pB 1 pHq, WOTq is unitary.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Invariant measures for frequently hypercyclic operators

Grivaux

Matheron

2014

Advances in Mathematics

View full text Add to dashboard Cite

Given a Polish topology τ on B1pXq, the set of all contraction operators on X " ℓp, 1 ď p ă 8 or X " c0, we prove several results related to the following question: does a typical T P B1pXq in the Baire Category sense has a non-trivial invariant subspace? In other words, is there a dense G δ set G Ď pB1pXq, τ q such that every T P G has a non-trivial invariant subspace? We mostly focus on the Strong Operator Topology and the StrongO perator Topology.

show abstract

Section: Proof Ofmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

Invariant measures for frequently hypercyclic operators

Grivaux

Matheron

2014

Advances in Mathematics

View full text Add to dashboard Cite

show abstract

“…Actually, when working in the whole of B M (H) or HC M (H), M > 1, we will always use SOT * rather than SOT. Indeed, with respect to SOT, a result from [21] gives a rather complete picture as far as typical properties are concerned: a typical element of (B 1 (H), SOT) is unitarily equivalent to the operator B (∞) , the countable direct ℓ 2 -sum of the unilateral backward shift B on ℓ 2 (N). It follows that for every M > 1, the class of operators T ∈ HC M (H) which are unitarily equivalent to M B (∞) is comeager in (HC M (H), SOT).…”

Section: Let (Tmentioning

confidence: 99%

“…This opens the way to Baire category arguments in B M (H) or HC M (H). Of course, there is nothing new in this observation: Baire category methods have already proved as useful in operator theory as anywhere else; see in particular [21].…”

mentioning

confidence: 99%

Linear dynamical systems on Hilbert spaces: typical properties and explicit examples

Grivaux¹,

Matheron²,

Menet³

2017

Preprint

View full text Add to dashboard Cite

We solve a number of questions pertaining to the dynamics of linear operators on Hilbert spaces, sometimes by using Baire category arguments and sometimes by constructing explicit examples. In particular, we prove the following results.(i) A typical hypercyclic operator is not topologically mixing, has no eigenvalues and admits no non-trivial invariant measure, but is densely distributionally chaotic. (ii) A typical upper-triangular operator is ergodic in the Gaussian sense, whereas a typical operator of the form "diagonal plus backward unilateral weighted shift" is ergodic but has only countably many unimodular eigenvalues; in particular, it is ergodic but not ergodic in the Gaussian sense. (iii) There exist Hilbert space operators which are chaotic and U-frequently hypercyclic but not frequently hypercyclic, Hilbert space operators which are chaotic and frequently hypercyclic but not ergodic, and Hilbert space operators which are chaotic and topologically mixing but not U-frequently hypercyclic. We complement our results by investigating the descriptive complexity of some natural classes of operators defined by dynamical properties.

show abstract

“…Various residuality results for the space of contractive C 0 -semigroups over a separable, infinite dimensional Hilbert space (under the topology of uniform weak convergence on compact subsets of R `) as well as for the space of contractions over a separable, infinite dimensional Hilbert space (under the pw-topology) have been investigated in [ES09], [Eis10], [EM10], etc. Residuality is only meaningful (viz.…”

Section: Introductionmentioning

confidence: 99%

The space of contractive $C_{0}$-semigroups is a Baire space

Dahya¹

2020

Preprint

View full text Add to dashboard Cite

Working over infinite dimensional separable Hilbert spaces, residual results have been achieved for the space of contractive C0-semigroups under the topology of uniform weak convergence on compact subsets of R`. Eisner, Mátrai, Serény, et al. raised in various publications 2008-10 the open problem: Does this space constitute a Baire space? Observing that the subspace of unitary semigroups is completely metrisable and appealing to known density results, we solve this problem by showing that certain topological properties can in general be transferred from dense subspaces to larger spaces.

show abstract

On typical properties of Hilbert space operators

Cited by 13 publications

References 31 publications

Invariant measures for frequently hypercyclic operators

Invariant measures for frequently hypercyclic operators

Linear dynamical systems on Hilbert spaces: typical properties and explicit examples

The space of contractive $C_{0}$-semigroups is a Baire space

Contact Info

Product

Resources

About