ANTISUPERCYCLIC OPERATORS AND ORBITS OF THE VOLTERRA OPERATOR

We say that a bounded linear operator T acting on a Banach space B is antisupercyclic if for any x ∈ B either Tnx = 0 for some positive integer n or the sequence {Tnx/ Tnx } 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 p ∞ and any f ∈ Lp[0, 1] then the limit limn→∞(n! V nf p)1/n does exist and equals 1 − inf supp (f). Upon using this asymptotic formula it is proved that the operator V acting on the Banach space Lp[0, 1] is antisupercyclic for any p ∈ (1,∞). The same statement for p = 1 or p = ∞ is false. The analogous results are proved for operators V zf(x) = (1/Γ(z)) 􀀀x 0 f(t)(x − t)z−1 dt when the real part of z ∈ C is positive.