Antisupercyclic Operators and Orbits of the Volterra Operator

We say that a bounded linear operator T acting on a Banach spaceB is antisupercyclic if for any x B either Tⁿx = 0 for somepositive integer n or the sequence {Tⁿx/||Tⁿx||} weakly convergesto zero in B. Antisupercyclicity of T means that the angle criterionof supercyclicity is not satisfied for T in the strongest possibleway. Normal antisupercyclic operators and antisupercyclic bilateralweighted shifts are characterized. As for the Volterra operator V, it is proved that if 1 p and any f L_p 0,1] then the limit lim_n (n!||Vⁿf||_p)^1/n doesexist and equals 1 – inf supp (f). Upon using this asymptoticformula it is proved that the operator V acting on the Banachspace L_p0,1] is antisupercyclic for any p (1,). The same statementfor p = 1 or p = is false. The analogous results are provedfor operators when the real part of z C is positive.