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Journal of the London Mathematical Society 2006 73(2):506-528; doi:10.1112/S0024610706022563
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© The London Mathematical Society

Antisupercyclic Operators and Orbits of the Volterra Operator

Stanislav Shkarin

King's College London, Department of Mathematics Strand, London, WC2R 2LS, United Kingdom stanislav.shkarin{at}kcl.ac.uk

Received 7 November 2004. Revision received 2 June 2005.

We say that a bounded linear operator T acting on a Banach space B is antisupercyclic if for any x isin 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 ≤ {infty} and any f isin Lp [0,1] then the limit limn->{infty} (n!||Vnf||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 isin (1,{infty}). The same statement for p = 1 or p = {infty} is false. The analogous results are proved for operators Formula when the real part of z isin C is positive.


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