turkmath.org

Linear dynamics, also known as hypercyclicity, examines the dynamics of linear operators on separable, infinite dimensional Banach spaces. Formally, an operator $T$ on a Banach space $X$ is hypercyclic if there exists a vector $x$ in $X$ for which its orbit $Orb(T, x) = \{x, T x, T^2x, T^3x, . . . \}$ is dense in $X$. Any such vector x is called a hypercyclic vector for the operator $T$. Over the years, the class of weighted shift operators has played an important role in hypercyclicity. They are often used when applying new results or as a testing ground for new ideas in hypercyclicity. In the present talk, we will discuss how weighted shift operators have contributed in the following three area of hypercyclicity: weak hypercyclicity, disjoint hypercyclicity, and common hypercyclic vectors.