Henrik Petersson

Hypercyclicity in omega

A sequence T = (Tn) of operators Tn:X → X is said to be hypercyclic if there exists a vector x ω X, called hypercyclic for T, such that {Tnx:n ≥ 0} is dense. A hypercyclic subspace for T is a closed infinitedimensional subspace of, except for zero, hypercyclic vectors. We prove that if T is a sequence of operators on. that has a hypercyclic subspace, then there exist (i) a sequence (pn) of one variable polynomials pn such that (pn). is hypercyclic for every fixed. and (ii) an operator S:→ that maps nonzero vectors onto hypercyclic vectors for T.

PDE-PRESERVING PROPERTIES

A continuous linear operator T, on the space of en- tire functions in d variables, is PDE-preserving for a given set P µC(»1;:::;»d) of polynomials if it maps every kernel-set kerP(D), P 2P, invariantly. It is clear that the set O(P) of PDE-preserving operators forP forms an algebra under composition. We study and link properties and structures on the operator side O(P) versus the corresponding familyP of polynomials. For our purposes, we intro- duce notions such as the PDE-preserving hull and basic sets for a given set P which, roughly, is the largest, respectively a minimal, collection of polynomials that generate all the PDE-preserving op- erators for P. We also describe PDE-preserving operators via a kernel theorem. We apply Hilberts Nullstellensatz.

HYPERCYCLICITY ON INVARIANT SUBSPACES

A continuous linear operator T : X -> X is called hypercyclic if there exists an x is an element of X such that the orbit {T(n)x}(n >= 0) is dense. We consider the problem: given an operator T : X -> X, hypercylic or not, is the restriction T vertical bar y to some closed invariant subspace Y subset of X hypercyclic? In particular, it is well-known that any non-constant partial differential operator p(D) on H(C-d) (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) : kerq(D) -> ker q(D) is hypercyclic. We also study hypercyclicity for other types of operators on subspaces of H (Cd).