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

Miihkinen, Santeri

doi:10.1090/proc/13180

Strict singularity of a Volterra-type integral operator on $H^p$
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by Santeri Miihkinen

Proc. Amer. Math. Soc. 145 (2017), 165-175

DOI: https://doi.org/10.1090/proc/13180

Published electronically: June 10, 2016

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Abstract:

We prove that the Volterra-type integral operator \[ T_gf(z) = \int _0^z f(\zeta )g’(\zeta )d\zeta , \quad z \in \mathbb {D},\] defined on the Hardy spaces $H^p$ fixes an isomorphic copy of $\ell ^p$ if it is not compact. In particular, the strict singularity of $T_g$ coincides with its compactness on spaces $H^p.$ As a consequence, we obtain a new proof for the equivalence of the compactness and the weak compactness of $T_g$ on $H^1$. Moreover, a non-compact $T_g$ acting on the space $BMOA$ fixes an isomorphic copy of $c_0.$

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