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


Author: Santeri Miihkinen
Journal: Proc. Amer. Math. Soc. 145 (2017), 165-175
MSC (2010): Primary 47G10; Secondary 30H10
DOI: https://doi.org/10.1090/proc/13180
Published electronically: June 10, 2016
MathSciNet review: 3565369
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Abstract: We prove that the Volterra-type integral operator

$\displaystyle T_gf(z) = \int _0^z f(\zeta )g'(\zeta )d\zeta , \quad z \in \mathbb{D},$

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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Additional Information

Santeri Miihkinen
Affiliation: Department of Mathematics and Statistics, University of Helsinki, Box 68, 00014 Helsinki, Finland
Email: santeri.miihkinen@helsinki.fi

DOI: https://doi.org/10.1090/proc/13180
Keywords: Volterra operator, integral operator, strict singularity, strictly singular, Hardy spaces
Received by editor(s): January 20, 2016
Received by editor(s) in revised form: March 1, 2016
Published electronically: June 10, 2016
Additional Notes: This research was supported by the Academy of Finland via the Centre of Excellence in Analysis and Dynamics Research
Communicated by: Pamela B. Gorkin
Article copyright: © Copyright 2016 American Mathematical Society