HYPERCYCLICITY OF OPERATORS THAT $$\lambda$$-COMMUTE WITH THE HARDY BACKWARD SHIFT

Abstract

Abstract An operator T acting on a separable complex Banach space $$\mathcal {B}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>B</mml:mi> </mml:math> is said to be hypercyclic if there exists $$f\in \mathcal {B}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>B</mml:mi> </mml:mrow> </mml:math> such that the orbit $$\{T^n f:\ n\in \mathbb {N}\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>{</mml:mo> <mml:msup> <mml:mi>T</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mspace/> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>N</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> is dense in $$\mathcal {B}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>B</mml:mi> </mml:math> . Godefroy and Shapiro (J. Funct. Anal., 98(2):229–269, 1991) characterized those elements, which are hypercyclic, in the commutant of the Hardy backward shift. In this paper, we study some dynamical properties of operators X that $$\lambda$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> </mml:math> -commute with the Hardy backward shift B , that is, $$BX=\lambda XB$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>B</mml:mi> <mml:mi>X</mml:mi> <mml:mo>=</mml:mo> <mml:mi>λ</mml:mi> <mml:mi>X</mml:mi> <mml:mi>B</mml:mi> </mml:mrow> </mml:math> .