Invariant Gaussian measures for operators on Banach spaces and linear dynamics

By Fr'ed'eric Bayart, S. Grivaux. 2007

94 citations recorded by the source corpus.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/82cc7a8c1dbd3b3fc065ae636aca5f8159c3092a

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Citations in the corpus

On a Theorem of Godefroy and Shapiro (2006, background): “For general Banach spaces, the criterion only seems to hold under additional assumptions, see [ 4 ].”
Recurrent Linear Operators (2013, background): “In fact, in this case, T is frequently hypercyclic; see [5,6,29,30].”
Distributional chaos for linear operators (2013, background): “Are there frequently hypercyclic operators which are not distributionally chaotic? For example, in [3] the authors constructed a frequently hypercyclic operator on c0 that is not Devaney chaotic (and not mixing).”
Mixing operators and small subsets of the circle (2011, methodology): “As shown in [8], this breaks down completely when p = ∞: there is a frequently hypercyclic weighted shift Bw on c0(N) whose weight sequence satisfies w1 · · ·wn = 1 for infinitely many n.”
Frequently recurrent operators (2020, background): “(d) By [4, Corollary 5.2] (see also [13, Theorem 7.3]) there exists a frequently hypercyclic weighted backward shift on c 0 ( N ) that is not chaotic.”
Upper frequent hypercyclicity and related notions (2016, background): “While, in general, chaos and frequent hypercyclicity are incomparable concepts (see [4], [28]), in the framework of weighted backward shifts on F-sequence spaces in which (en) is an unconditional basis, chaos implies frequent hypercyclicity, see [19, Corollary 9.”
Szemeredi's theorem, frequent hypercyclicity and multiple recurrence (2010, background): “h l2(N), in c 0(N) there exists a unilateral weighted shift which is frequently hypercyclic and thus topologically multiply recurrent but is neither chaotic nor mixing. Such an example is provided in [3]. Proposition 4.10. There exists a hypercyclic bilateral weighted shift on l2(Z) which is not top”
Difference sets and frequently hypercyclic weighted shifts (2013, background): “This notion has then been investigated by several authors; see, for instance, [3, 5, 8, 10, 11, 16].”
Linear dynamical systems on Hilbert spaces: typical properties and explicit examples (2017, background): “Properties of unimodular eigenvectors of an operator T ∈ B(H) are closely related to those of invariant Gaussian measures for T (see for instance [5], [6], [8]): indeed, it turns out that in fact SPAN(H) = G-INVf (H) and PSPAN(H) = G-ERG(H); so that there was in fact no need for introducing new nota”
Linear chaos and frequent hypercyclicity (2014, methodology): “[4] F. Bayart and S. Grivaux, Frequently hypercyclic operators, Trans.”
Hilbertian Jamison sequences and rigid dynamical systems (2011, background): “See [5] for extensions to the Banach space setting, and the book [6, Ch.”
Invariant measures for frequently hypercyclic operators (2014, background): “The quest for similar results for operators acting on general Banach spaces was begun in [3] and culminated in [5], where a very general condition (valid on any complex Banach space X) for an operator to admit a Gaussian ergodic measure with full support was obtained.”
Linear subsets of nonlinear sets in topological vector spaces (2013, background): “1, composition operators Cφ on H(D) with φ an automorphism without fixed points, nonscalar differential operators Φ(D) on H(C), and the Rolewicz operator λB with |λ| > 1 on c0 or p, 1 ≤ p < ∞, among others) are also frequently hypercyclic; see [47], [48], [103], and [104].”