ANTISUPERCYCLIC OPERATORS AND ORBITS OF THE VOLTERRA OPERATOR

Abstract

We say that a bounded linear operator T acting on a Banach space B is antisupercyclic if for any $x\in B$ either $T^nx=0$ for some positive integer n or the sequence $\{T^nx/\|T^nx\|\}$ weakly converges to zero in B. Antisupercyclicity of T means that the angle criterion of supercyclicity is not satisfied for T in the strongest possible way. Normal antisupercyclic operators and antisupercyclic bilateral weighted shifts are characterized.

As for the Volterra operator V, it is proved that if $1\leq p\leq\infty$ and any $f\in L_p[0,1]$ then the limit $\lim_{n\to\infty} (n!\|V^nf\|_p)^{1/n}$ does exist and equals $1-\inf\,\hbox{\rm supp}\,(f)$. Upon using this asymptotic formula it is proved that the operator V acting on the Banach space $L_p[0,1]$ is antisupercyclic for any $p\in(1,\infty)$. The same statement for $p=1$ or $p=\infty$ is false. The analogous results are proved for operators $V^zf(x)=(1/{\Gamma(z)})\int_0^x f(t) (x-t)^{z-1}\,dt$ when the real part of $z\in{\mathbb C}$ is positive.