Supercyclic Properties of Extended Eigenoperators of the Differentiation Operator on the Space of Entire Functions

Abstract A continuous linear operator L defined on the space of entire functions $$\mathcal {H}({\mathbb {C}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>(</mml:mo> <mml:mi>C</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is said to be an extended $$\lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> </mml:math> -eigenoperator of the differentiation operator D provided $$DL=\lambda LD$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mi>L</mml:mi> <mml:mo>=</mml:mo> <mml:mi>λ</mml:mi> <mml:mi>L</mml:mi> <mml:mi>D</mml:mi> </mml:mrow> </mml:math> . Here we fully characterize when an extended $$\lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> </mml:math> -eigenoperator of D is supercyclic, it has a hypercyclic subspace or it has a supercyclic subspace.