scholarly journals Topological Transitivity of Shift Similar Operators on Nonseparable Hilbert Spaces

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Andriy Zagorodnyuk ◽  
Zoriana Novosad
Keyword(s):  
Hilbert Spaces ◽  
Differential Operators ◽  
Weighted Shift ◽  
Weighted Shifts ◽  
Topological Transitivity ◽  
Shift Operators

In this paper, we investigate topological transitivity of operators on nonseparable Hilbert spaces which are similar to backward weighted shifts. In particular, we show that abstract differential operators and dual operators to operators of multiplication in graded Hilbert spaces are similar to backward weighted shift operators.

Existence Theorems for Some Weak Abstract Variable Domain Hyperbolic Problems

1971 ◽  
Vol 23 (4) ◽  
pp. 611-626 ◽  
Author(s):  
Robert Carroll ◽  
Emile State
Keyword(s):  
Hilbert Spaces ◽  
Differential Operators ◽  
Separable Hilbert Space ◽  
Hyperbolic Equations ◽  
Existence Theorems ◽  
Hyperbolic Problems ◽  
Linear Maps ◽  
Variable Domains ◽  
Elliptic Differential Operators ◽  
Image Position

In this paper we prove some existence theorems for some weak problems with variable domains arising from hyperbolic equations of the type1.1where A = {A(t)} is, for example, a family of elliptic differential operators in space variables x = (x1, …, xn). Thus let H be a separable Hilbert space and let V(t) ⊂ H be a family of Hilbert spaces dense in H with continuous injections i(t): V(t) → H (0 ≦ t ≦ T < ∞). Let V’ (t) be the antidual of V(t) (i.e. the space of continuous conjugate linear maps V(t) → C) and using standard identifications one writes V(t) ⊂ H ⊂ V‘(t).

CHAOS IN A CLASS OF NONCONSTANT WEIGHTED SHIFT OPERATORS

2013 ◽  
Vol 23 (01) ◽  
pp. 1350010 ◽  
Author(s):  
XINXING WU ◽  
PEIYONG ZHU
Keyword(s):  
Positive Integer ◽  
Linear Space ◽  
Normed Linear Space ◽  
Shift Operator ◽  
Weighted Shift ◽  
Constructive Proof ◽  
Sufficient Condition ◽  
Weighted Shift Operator ◽  
Shift Operators ◽  
Devaney Chaos

In this paper, chaos generated by a class of nonconstant weighted shift operators is studied. First, we prove that for the weighted shift operator Bμ : Σ(X) → Σ(X) defined by Bμ(x0, x1, …) = (μ(0)x1, μ(1)x2, …), where X is a normed linear space (not necessarily complete), weak mix, transitivity (hypercyclity) and Devaney chaos are all equivalent to separability of X and this property is preserved under iterations. Then we get that [Formula: see text] is distributionally chaotic and Li–Yorke sensitive for each positive integer N. Meanwhile, a sufficient condition ensuring that a point is k-scrambled for all integers k > 0 is obtained. By using these results, a simple example is given to show that Corollary 3.3 in [Fu & You, 2009] does not hold. Besides, it is proved that the constructive proof of Theorem 4.3 in [Fu & You, 2009] is not correct.

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