# Strict singularity of a Volterra-type integral operator on H^p

#### Description

We consider a Volterra-type integral operator $$T_gf(z)=\int_0^z f(\zeta)g'(\zeta) d\zeta,$$ acting on the Hardy spaces H^p of the unit disc. The operator T_g was introduced by Ch. Pommerenke and it has been studied systematically by several people including A. Aleman, A.G. Siskakis and R. Zhao among others. From a functional analytic point of view, one interesting notion is the strict singularity of a linear operator between Banach spaces. An operator is strictly singular if its restriction to any infinite-dimensional subspace is not an isomorphism onto its range. We discuss our recent result, which states that a non-compact T_g fixes an isomorphic copy of the sequence space l^p. In particular, the strict singularity of T_g coincides with its compactness on spaces H^p.

*This paper has been withdrawn.*

Strict singularity of a Volterra-type integral operator on H^p

We consider a Volterra-type integral operator $$T_gf(z)=\int_0^z f(\zeta)g'(\zeta) d\zeta,$$ acting on the Hardy spaces H^p of the unit disc. The operator T_g was introduced by Ch. Pommerenke and it has been studied systematically by several people including A. Aleman, A.G. Siskakis and R. Zhao among others. From a functional analytic point of view, one interesting notion is the strict singularity of a linear operator between Banach spaces. An operator is strictly singular if its restriction to any infinite-dimensional subspace is not an isomorphism onto its range. We discuss our recent result, which states that a non-compact T_g fixes an isomorphic copy of the sequence space l^p. In particular, the strict singularity of T_g coincides with its compactness on spaces H^p.