Theoretical Statistics and Mathematics Unit, ISI Delhi

September 9, 2015 (Wednesday) ,
3:30 PM at Webinar

Speaker:
Aneesh M,
Indian Institute of Technology Kanpur

Title:
Hypercyclicity and Frequent Hypercyclicity

Abstract of Talk

An operator $T$ on a separable Banach space $X$ is called hypercyclic if there exists an element $x\in X$ such that $\{x,Tx,T^2x,...\}$ is dense in $X$. Several operators including weighted shifts on $\ell^p$ and adjoints of multiplication operators on the Hardy space are hypercyclic. Hypercyclicity has been studied extensively for the last three decades in the connection with the invariant subset problem and topological dynamics. In 2006 F. Bayart and S. Grivaux further strengthened this concept to frequent hypercyclicity: an operator $T$ on $X$ is said to be frequently hypercyclic if there exists an $x\in X$ such that $\{x,Tx,T^2x,...\}$ is frequently dense. We first present an introduction to hypercyclic and frequently hypercyclic operators. Then we look at the hypercyclicity and frequent hypercyclicity of the left multiplication operator $L_T(S)=TS$ and the conjugate operator $C_T(S)=TST^*$ on certain spaces of operators defined on a Hilbert space.

Some of the results are based on a joint work with Prof. Manjul Gupta.