[Banach] Abstract of a paper by Santeri Miihkinen

This is an announcement for the paper "Strict singularity of a Volterra-type integral operator on $H^p$" by Santeri Miihkinen.

Abstract: We prove that a Volterra-type integral operator $T_gf(z) = \int_0^z f(\zeta)g'(\zeta)d\zeta, , z \in \mathbb D,$ defined on Hardy spaces $H^p, , 1 \le p < \infty,$ fixes an isomorphic copy of $\ell^p,$ if the operator $T_g$ is not compact. In particular, this shows that the strict singularity of the operator $T_g$ coincides with the compactness of the operator $T_g$ on spaces $H^p.$ As a consequence, we obtain a new proof for the equivalence of the compactness and the weak compactness of the operator $T_g$ on $H^1$.

Archive classification: math.FA

Mathematics Subject Classification: 47G10 (Primary) 30H10 (Secondary )

Remarks: 14 pages, 1 figure

Submitted from: santeri.miihkinen@helsinki.fi

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1509.08356

or

http://arXiv.org/abs/1509.08356