scholarly journals Disjoint supercyclic powers of weighted shifts on weighted sequence spaces

2014 ◽  
Vol 38 ◽  
pp. 1007-1022 ◽  
Author(s):  
Yu-Xia LIANG ◽  
Ze-Hua ZHOU
Keyword(s):  
Sequence Spaces ◽  
Weighted Shifts ◽  
Weighted Sequence

BACKWARD 3-STEP EXTENSIONS OF RECURSIVELY GENERATED WEIGHTED SHIFTS: A RANGE OF QUADRATIC HYPONORMALITY

2013 ◽  
Vol 89 (3) ◽  
pp. 488-493
Author(s):  
GEORGE R. EXNER ◽  
IL BONG JUNG ◽  
MI RYEONG LEE ◽  
SUN HYUN PARK
Keyword(s):  
Open Problem ◽  
Weighted Shift ◽  
Real Numbers ◽  
Weighted Shifts ◽  
Positive Real ◽  
Weight Sequence ◽  
Weighted Sequence ◽  
Quadratically Hyponormal

AbstractLet $\alpha : 1, 1, \sqrt{x} , \mathop{( \sqrt{u} , \sqrt{v} , \sqrt{w} )}\nolimits ^{\wedge } $ be a backward 3-step extension of a recursively generated weighted sequence of positive real numbers with $1\leq x\leq u\leq v\leq w$ and let ${W}_{\alpha } $ be the associated weighted shift with weight sequence $\alpha $. The set of positive real numbers $x$ such that ${W}_{\alpha } $ is quadratically hyponormal for some $u, v$ and $w$ is described, solving an open problem due to Curto and Jung [‘Quadratically hyponormal weighted shifts with two equal weights’, Integr. Equ. Oper. Theory 37 (2000), 208–231].

scholarly journals Chaos and frequent hypercyclicity for weighted shifts

2020 ◽  
pp. 1-37
Author(s):  
STÉPHANE CHARPENTIER ◽  
KARL GROSSE-ERDMANN ◽  
QUENTIN MENET
Keyword(s):  
Difference Sets ◽  
Weighted Shift ◽  
Sequence Spaces ◽  
Hypercyclic Operators ◽  
Weighted Shifts ◽  
Banach Sequence Spaces ◽  
Frequently Hypercyclic Operators ◽  
Köthe Sequence Spaces ◽  
The Relationship ◽  
Banach Sequence

Abstract Bayart and Ruzsa [Difference sets and frequently hypercyclic weighted shifts. Ergod. Th. & Dynam. Sys.35 (2015), 691–709] have recently shown that every frequently hypercyclic weighted shift on $\ell ^p$ is chaotic. This contrasts with an earlier result of Bayart and Grivaux [Frequently hypercyclic operators. Trans. Amer. Math. Soc.358 (2006), 5083–5117], who constructed a non-chaotic frequently hypercyclic weighted shift on $c_0$ . We first generalize the Bayart–Ruzsa theorem to all Banach sequence spaces in which the unit sequences form a boundedly complete unconditional basis. We then study the relationship between frequent hypercyclicity and chaos for weighted shifts on Fréchet sequence spaces, in particular, on Köthe sequence spaces, and then on the special class of power series spaces. We obtain, rather curiously, that every frequently hypercyclic weighted shift on $H(\mathbb {D})$ is chaotic, while $H(\mathbb {C})$ admits a non-chaotic frequently hypercyclic weighted shift.

scholarly journals FREQUENTLY HYPERCYCLIC BILATERAL SHIFTS

2018 ◽  
Vol 61 (2) ◽  
pp. 271-286 ◽  
Author(s):  
KARL-G. GROSSE-ERDMANN
Keyword(s):  
Positive Answer ◽  
Weighted Shift ◽  
Sequence Spaces ◽  
Weighted Shifts ◽  
Banach Sequence Spaces ◽  
Bilateral Weighted Shift ◽  
Banach Sequence

AbstractIt is not known whether the inverse of a frequently hypercyclic bilateral weighted shift on c0(ℤ) is again frequently hypercyclic. We show that the corresponding problem for upper frequent hypercyclicity has a positive answer. We characterise, more generally, when bilateral weighted shifts on Banach sequence spaces are (upper) frequently hypercyclic.

scholarly journals Lower bounds for matrices on block weighted sequence spaces I

2009 ◽  
Vol 59 (1) ◽  
pp. 81-94 ◽  
Author(s):  
R. Lashkaripour ◽  
D. Foroutannia
Keyword(s):  
Lower Bounds ◽  
Sequence Spaces ◽  
Weighted Sequence
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