Chaotic weighted shifts in Bargmann space
The authors study the chaotic behaviour of unbounded weighted backward shifts on the Bargmann space \(F(\mathbb{C})\), that is, the set of entire functions which can be written as \(f(z)=\sum_{k\geq 0} c_{k} \frac {z^{k}}{\sqrt{k!}}\), where \(\sum_{k\geq 0} | c_{k}| ^{2}\) is finite. If \(T\) is the backward shift with weights \((\omega _{k})_{k\geq 0}\) defined by \[ T\left(\sum_{k\geq 0} c_{k} \frac {z^{k}}{\sqrt{k!}}\right)= \sum_{k\geq 0} \omega _{k}c_{k+1} \frac {z^{k}}{\sqrt{k!}}, \] it is proved that \(T\) is chaotic if and only if the series \[ \sum_{n=1}^{\infty } \prod_{j=0}^{n-1}\frac {1}{| \omega _{j}| ^{2}} \] is convergent. In particular, the annihilation operator \(A: f \longmapsto f'\) is chaotic on \(F(\mathbb{C})\).
scientific article; zbMATH DE number 2184068
scientific article