Hypercyclic operators on spaces of holomorphic functions

SAVRANSKY MARTIN

CAPITAL FEDERAL

Congreso; WIDABA 2014; 2014

UNIVERSIDAD TORCUATO DI TELLA

A theorem of G. Godefroy and J. H. Shapiro [2] states that every operator on H(C n ), n ≥ 1, that commutes with the translations τ a f(z) = f(z+a), a ∈ C n , and it is not a scalar multiple of the identity is hypercyclic. This family of operatos is known as convolution operators. In the literature, there aren?t many examples of hypercyclic non-convolution operators on H(C n ). R. Aron and D. Markose [1] studied hypercyclic properties of the operator Tf(z) = f 0 (λz +b), for λ,b ∈ C, which is not a convolution operator if λ 6= 1. They prove that T is hypercyclic if |λ| ≥ 1, and that T is not hypercyclic if |λ| < 1 and b = 0. In this talk, we will comment hypercyclic properties of operators defined on spaces of holomorphic functions on C n and on complex Banach spaces with unconditional basis. We will consider operators such as Tf(z) = D α f(λz + b), that is, operators which are a composition between a differentiation operator and a composition one. Here α indicates ?the amount? of derivatives in the canonical directions, and the symbol of the composition operator, z 7→ λz + b, is a diagonal affine application. As the dimension increases, new difficulties appear and the hypercyclicity of T does not only depends on the size of λ as in the one dimensional case.