Abstract
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.
Highlights
Let X be a Hausdorff locally convex space
A continuous linear operator T : X ⟶ X acting on a separable Fréchet space X is called hypercyclic if there is a vector x ∈ X for which the orbit under T, OrbðT
For the space of all entire functions, Birkhoff [2] proved that the translation operator Ta : f ðxÞ ↦ f ðx + aÞ, ða ≠ 0Þ is hypercyclic
Summary
Let X be a Hausdorff locally convex space. A continuous operator T : X ⟶ X is called topologically transitive if for each pair U,V of nonempty open subsets of X, there is some n ∈ N with TnðUÞ ∩ V ≠ ∅: If the underlying space is a separable Baire space, the transitivity is equivalent to the hypercyclicity by Birkhoff’s transitivity theorem (see [1], p. 2). Note that in [7], item (iii) is written: “The restriction T : Y ⟶ Y to any T-invariant (separable) closed subspace Y ⊂ l2ðHÞ is hypercyclic.” It is not correct because the subspace Y0 consisting of vectors ðx0, 0, 0, ⋯Þ,x0 ∈ H is invariant but the restriction of T to any separable subspace of Y0 is not hypercyclic since Y0 ⊂ ker T: In Section 2, we consider abstract shift similar operators on nonseparable function Hilbert spaces l2ðHÞ: In particular, abstract differentiation operators and dual operators to abstract multiplication operators can be considered as abstract shift similar operators.
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