Hypercyclicity on invariant subspaces
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A continuous linear operator T : X → X is called hypercyclic if there exists an x ∈ X such that the orbit {Tnx}n≥0 is dense. We consider the problem: given an operator T : X → X, hypercyclic or not, is the restriction TjY to some closed invariant subspace Y ⊂ X hypercyclic? In particular, it is well-known that any non-constant partial differential operator p(D) on H(Cd) (entire functions) is hypercyclic. Now, if q(D) is another such operator, p(D) maps ker q(D) invariantly (by commutativity), and we obtain a necessary and sufficient condition on p and q in order that the restriction p(D) : ker q(D) → ker q(D) is hypercyclic. We also study hypercyclicity for other types of operators on subspaces of H(Cd).
Abstract
A continuous linear operator T : X → X is called hypercyclic if there exists an x ∈ X such that the orbit {Tnx}n≥0 is dense. We consider the problem: given an operator T : X → X, hypercyclic or not, is the restriction TjY to some closed invariant subspace Y ⊂ X hypercyclic? In particular, it is well-known that any non-constant partial differential operator p(D) on H(Cd) (entire functions) is hypercyclic. Now, if q(D) is another such operator, p(D) maps ker q(D) invariantly (by commutativity), and we obtain a necessary and sufficient condition on p and q in order that the restriction p(D) : ker q(D) → ker q(D) is hypercyclic. We also study hypercyclicity for other types of operators on subspaces of H(Cd).
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