Strict singularity of Volterra type operators on Hardy spaces

Abstract

In this paper, we first characterize the boundedness and compactness of Volterra type operator Sgf(z)=∫0zf′(ζ)g(ζ)dζ,z∈D, defined on Hardy spaces Hp,0<p<∞. The spectrum of Sg is also obtained. Then we prove that Sg fixes an isomorphic copy of ℓp and an isomorphic copy of ℓ2 if the operator Sg is not compact on Hp(1≤p<∞). In particular, this implies that the strict singularity of the operator Sg coincides with the compactness of the operator Sg on Hp. At last, we post an open question for further study.