Frequently hypercyclic bilateral shifts

Transcription

1 Frequently hypercyclic bilateral shifts Karl Grosse-Erdmann Département de Mathématique Université de Mons, Belgium October 19, 2017 Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 1 / 18

2 How not to solve a problem that is not due to Bayart and Grivaux Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 2 / 18

3 Linear dynamics In linear dynamics, the three central notions are those of hypercyclic operators chaotic operators frequently hypercyclic operators Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 3 / 18

4 An operator T : X X on a separable Banach space X is hypercyclic if there is some x X (called hypercyclic vector) such that orb(x, T ) = {x, Tx, T 2 x,...} is dense in X. Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 4 / 18

5 An operator T : X X on a separable Banach space X is hypercyclic if there is some x X (called hypercyclic vector) such that orb(x, T ) = {x, Tx, T 2 x,...} is dense in X. By the Birkho transitivity theorem (1920), an operator T is hypercyclic if and only if U, V open, n : T n (U) V. Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 4 / 18

6 An operator T : X X on a separable Banach space X is hypercyclic if there is some x X (called hypercyclic vector) such that orb(x, T ) = {x, Tx, T 2 x,...} is dense in X. By the Birkho transitivity theorem (1920), an operator T is hypercyclic if and only if U, V open, n : T n (U) V. And then the set of hypercyclic vectors is residual. Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 4 / 18

7 An operator T : X X on a separable Banach space X is hypercyclic if there is some x X (called hypercyclic vector) such that orb(x, T ) = {x, Tx, T 2 x,...} is dense in X. By the Birkho transitivity theorem (1920), an operator T is hypercyclic if and only if U, V open, n : T n (U) V. And then the set of hypercyclic vectors is residual. All of this thanks to Baire's theorem. Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 4 / 18

8 Now, T n (U) V U T n (V ). Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 5 / 18

9 Now, T n (U) V U T n (V ). Corollary If T is invertible, then T hypercyclic = T 1 hypercyclic. This is a consequence of Baire. Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 5 / 18

10 Now, T n (U) V U T n (V ). Corollary If T is invertible, then T hypercyclic = T 1 hypercyclic. This is a consequence of Baire. In fact, the result fails for operators on normed spaces. Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 5 / 18

11 Chaos Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 6 / 18

12 Chaos An operator T on a separable Banach space X is chaotic if it is hypercyclic and it has a dense set of periodic points. Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 6 / 18

13 Chaos An operator T on a separable Banach space X is chaotic if it is hypercyclic and it has a dense set of periodic points. Now, T and T 1 have the same periodic points. Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 6 / 18

14 Chaos An operator T on a separable Banach space X is chaotic if it is hypercyclic and it has a dense set of periodic points. Now, T and T 1 have the same periodic points. Corollary If T is invertible, then Another triviality... T chaotic = T 1 chaotic. Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 6 / 18

15 Frequent hypercyclicity Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 7 / 18

16 Frequent hypercyclicity Denition (Bayart, Grivaux 2004) An operator T : X X is called frequently hypercyclic if there is some x X (called frequently hypercyclic vector) such that U open, dens{n 0 ; T n x U} > 0. Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 7 / 18

17 Frequent hypercyclicity Denition (Bayart, Grivaux 2004) An operator T : X X is called frequently hypercyclic if there is some x X (called frequently hypercyclic vector) such that Recall that, for A N 0, U open, dens{n 0 ; T n x U} > 0. dens A = lim inf N #{n N : n A}. N + 1 Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 7 / 18

18 Now things get interesting: Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 8 / 18

19 Now things get interesting: The set of frequently hypercyclic vectors is often (Bayart-Grivaux, 2006), in fact always (Moothathu 2013, Bayart-Ruzsa 2015) of rst Baire category, hence non-residual. Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 8 / 18

20 Now things get interesting: The set of frequently hypercyclic vectors is often (Bayart-Grivaux, 2006), in fact always (Moothathu 2013, Bayart-Ruzsa 2015) of rst Baire category, hence non-residual. Thus, Baire's theorem is of no use for proving that an operator is frequently hypercyclicity. Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 8 / 18

21 Now things get interesting: The set of frequently hypercyclic vectors is often (Bayart-Grivaux, 2006), in fact always (Moothathu 2013, Bayart-Ruzsa 2015) of rst Baire category, hence non-residual. Thus, Baire's theorem is of no use for proving that an operator is frequently hypercyclicity. Problem (Bayart, Grivaux 2006) If T is invertible, do we have T frequently hypercyclic = T 1 frequently hypercyclic? Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 8 / 18

22 Now things get interesting: The set of frequently hypercyclic vectors is often (Bayart-Grivaux, 2006), in fact always (Moothathu 2013, Bayart-Ruzsa 2015) of rst Baire category, hence non-residual. Thus, Baire's theorem is of no use for proving that an operator is frequently hypercyclicity. Problem (Bayart, Grivaux 2006) If T is invertible, do we have T frequently hypercyclic = T 1 frequently hypercyclic? This is Problem 44 in A. J. Guirao, V. Montesinos, V. Zizler: Open problems in the geometry and analysis of Banach spaces (2016). Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 8 / 18

23 There seems to be consensus that the problem of Bayart-Grivaux should have a negative solution. Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 9 / 18

24 There seems to be consensus that the problem of Bayart-Grivaux should have a negative solution. It is natural to look for bilateral weighted shifts as possible candidates, that is, operators of the form with weights w n > 0. B w (x n ) n Z = (w n+1 x n+1 ) n Z Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 9 / 18

25 There seems to be consensus that the problem of Bayart-Grivaux should have a negative solution. It is natural to look for bilateral weighted shifts as possible candidates, that is, operators of the form B w (x n ) n Z = (w n+1 x n+1 ) n Z with weights w n > 0. However: Theorem (Bayart, Ruzsa 2015) On l p (Z), 1 p <, B w frequently hypercycylic B w chaotic. Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 9 / 18

26 There seems to be consensus that the problem of Bayart-Grivaux should have a negative solution. It is natural to look for bilateral weighted shifts as possible candidates, that is, operators of the form with weights w n > 0. However: Theorem (Bayart, Ruzsa 2015) On l p (Z), 1 p <, B w (x n ) n Z = (w n+1 x n+1 ) n Z B w frequently hypercycylic B w chaotic. This implies that Bayart-Grivaux has a positive solution for the B w on l p (Z). Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 9 / 18

27 How about B w on c 0 (Z)? Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 10 / 18

28 How about B w on c 0 (Z)? Theorem (Bayart, Ruzsa 2015) An invertible B w is frequently hypercyclic on c 0 (Z) there is a sequence (A p ) p 1 of pairwise disjoint sets of positive lower density such that (sym) p, q 1, min n Ap,m Aq n m > p + q; n m p, q 1, m A q, n A p : w n m+1 w 0 < 1 2 p+q (n < m), w 1 w n m > 2 p+q (n < m). (non-sym) p 1, w1 w n as n, n A p. Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 10 / 18

29 How about B w on c 0 (Z)? Theorem (Bayart, Ruzsa 2015) An invertible B w is frequently hypercyclic on c 0 (Z) there is a sequence (A p ) p 1 of pairwise disjoint sets of positive lower density such that (sym) p, q 1, min n Ap,m Aq n m > p + q; n m p, q 1, m A q, n A p : w n m+1 w 0 < 1 2 p+q (n < m), w 1 w n m > 2 p+q (n < m). (non-sym) p 1, w1 w n as n, n A p. So we also know when B 1 w is frequently hypercyclic on c 0(Z)! Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 10 / 18

30 How about B w on c 0 (Z)? Theorem (Bayart, Ruzsa 2015) An invertible B w is frequently hypercyclic on c 0 (Z) there is a sequence (A p ) p 1 of pairwise disjoint sets of positive lower density such that (sym) p, q 1, min n Ap,m Aq n m > p + q; n m p, q 1, m A q, n A p : w n m+1 w 0 < 1 2 p+q (n < m), w 1 w n m > 2 p+q (n < m). (non-sym) p 1, w1 w n as n, n A p. So we also know when B 1 w is frequently hypercyclic on c 0(Z)! (my) Conjecture There is an invertible weighted backward shift B w on c 0 (Z) that is frequently hypercyclic, but its inverse is not. Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 10 / 18

31 Upper frequent hypercyclicity There is a poor man's version of frequent hypercyclicity: replace lower by upper density. Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 11 / 18

32 Upper frequent hypercyclicity There is a poor man's version of frequent hypercyclicity: replace lower by upper density. Denition (Shkarin 2009) An operator T : X X is called upper frequently hypercyclic if there is some x X (called upper frequently hypercyclic vector) such that U open, dens{n 0 ; T n x U} > 0. Here, of course, dens A = lim sup N #{n N:n A} N+1. Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 11 / 18

33 Upper frequent hypercyclicity There is a poor man's version of frequent hypercyclicity: replace lower by upper density. Denition (Shkarin 2009) An operator T : X X is called upper frequently hypercyclic if there is some x X (called upper frequently hypercyclic vector) such that U open, dens{n 0 ; T n x U} > 0. Here, of course, dens A = lim sup N #{n N:n A} N+1. But upper frequent hypercyclicity is more interesting than it seems! Theorem (Bayart, Ruzsa 2015) If T is upper frequently hypercyclic then the set of upper frequently hypercyclic vectors for T is residual. Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 11 / 18

34 So, Baire is back! Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 12 / 18

35 So, Baire is back! Theorem (Bonilla, G-E, 2016) Let T be an operator. The following assertions are equivalent: (a) (b) T is upper frequently hypercyclic; V open, δ > 0, U open, x U dens {n 0 : T n x V } > δ. The proof uses, naturally, Baire's theorem. Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 12 / 18

36 So, Baire is back! Theorem (Bonilla, G-E, 2016) Let T be an operator. The following assertions are equivalent: (a) (b) T is upper frequently hypercyclic; V open, δ > 0, U open, x U dens {n 0 : T n x V } > δ. The proof uses, naturally, Baire's theorem. But the symmetry of the Birkho transitivity theorem is lost. So we again have the following. Problem If T is invertible, do we have T upper frequently hypercyclic = T 1 upper frequently hypercyclic? Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 12 / 18

37 It is not clear (to me) if the upper problem should have a positive or a negative solution. Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 13 / 18

38 It is not clear (to me) if the upper problem should have a positive or a negative solution. Again, one would rst look at weighted backward shifts B w. Theorem (Bayart, Ruzsa 2015) On l p (Z), 1 p <, B w upper frequently hypercycylic B w chaotic. So the problem has a positive solution for the B w on l p (Z). Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 13 / 18

39 How about c 0 (Z)? Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 14 / 18

40 How about c 0 (Z)? Theorem (G-E 2017) If B w be an invertible weighted backward shift on c 0 (Z). Then B w upper frequently hypercyclic = B 1 w upper frequently hypercyclic. Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 14 / 18

41 How about c 0 (Z)? Theorem (G-E 2017) If B w be an invertible weighted backward shift on c 0 (Z). Then B w upper frequently hypercyclic = B 1 w upper frequently hypercyclic. Thus, for upper frequent hypercyclicity one has to look for other underlying spaces or other operators to construct counter-examples (if there are any). Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 14 / 18

42 Idea of the proof The proof contains a constructive part and a Baire argument. Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 15 / 18

43 Idea of the proof Theorem (Bayart, Ruzsa 2015) An invertible B w is upper frequently hypercyclic on c 0 (Z) there is a sequence (A p ) p 1 of pairwise disjoint sets of positive upper density such that (sym) p, q 1, min n Ap,m Aq n m > p + q; n m p, q 1, m A q, n A p : w n m+1 w 0 < 1 2 p+q (n < m), w 1 w n m > 2 p+q (n < m). (non-sym) p 1, w1 w n as n, n A p. Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 15 / 18

44 Idea of the proof Proposition 1 Let B w be invertible on c 0 (Z). TFAE: (a) there is a sequence (A p ) p 1 of pairwise disjoint sets of positive upper density such that (sym) p, q 1, min n Ap,m Aq n m > p + q; n m p, q 1, m A q, n A p : w n m+1 w 0 < 1 2 p+q (n < m), w 1 w n m > 2 p+q (n < m). (b) B w : l (Z) l (Z) is upper frequently hypercyclic for c 0 (Z). The proof is constructive. Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 15 / 18

45 Proposition 2 Let B w be invertible on c 0 (Z). TFAE: (b) B w : l (Z) l (Z) is upper frequently hypercyclic for c 0 (Z). (c) B w : c 0 (Z) c 0 (Z) is upper frequently hypercyclic. The proof uses Baire. Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 16 / 18

46 Proposition 2 Let B w be invertible on c 0 (Z). TFAE: (b) B w : l (Z) l (Z) is upper frequently hypercyclic for c 0 (Z). (c) B w : c 0 (Z) c 0 (Z) is upper frequently hypercyclic. The proof uses Baire. Thus, the upper frequent hypercyclicity of B w on c 0 (Z) is characterized by (Sym), which implies that Bw 1 is also upper frequently hypercyclic. Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 16 / 18

47 Thank you! Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 17 / 18

48 F. Bayart and S. Grivaux, Frequently hypercyclic operators, Trans. Amer. Math. Soc. 358 (2006), F. Bayart and I. Z. Ruzsa, Dierence sets and frequently hypercyclic weighted shifts, Ergodic Theory Dynam. Systems 35 (2015), A. Bonilla and K.-G. Grosse-Erdmann, Upper frequent hypercyclicity and related notions, arxiv: K.-G. Grosse-Erdmann, Frequently hypercyclic bilateral shifts, arxiv: A. J. Guirao, V. Montesinos, and V. Zizler, Open problems in the geometry and analysis of Banach spaces, Springer, [Cham] Karl Grosse-Erdmann (UMons) Frequently hypercyclic bilateral shifts NoLiFA 18 / 18