School of Mathematical and Computing Sciences

Komplexa seminariet (The Complex Analysis Seminar)

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Henrik Petersson: Approximation of homogeneous solutions to PDE:s via Hypercyclicity, continued

Abstract: A continuous linear operator T on a locally convex space X is said to be Hypercyclic if there exists a vector x in X, called hypercyclic for T, such that the orbit {x,Tx,TTx,...} is dense. A well-known theorem states that any nonconstant partial differential operator P(D) with constant coefficients, is hypercyclic on the Frechet space H(C^n) of n-variable entire functions. We note now, by commutativity, that if Q(D) is any other such differential operator, P(D) maps Ker Q(D) invariantly. This suggests studying conditions on P(D) and Q(D) in order that the restriction of P(D) to Ker Q(D) is hypercyclic. Some weeks ago we proposed this problem and presented some preliminary results. In this talk we present some recent progress. We prove the following:

THM: If Q is a polynomial with distinct irreducible factors Q', then P(D) is hypercyclic on Ker Q(D) provided the restriction of P to every irreducible component Z(Q') of Z(Q) is nonconstant.

The talk is independent from the first talk.

Onsdag 1 mars kl 10.15, Mallvinden

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