Abstract
A Banach space operator \(T\) is said to be weakly super convex-cyclic if there exists \(x \in X\) such that \(\{\lambda p(T )x : p\, \mathrm{convex \,polynomial}, \lambda \in \mathbb {C}\}\) is weakly dense in \(X\). The notion of convex-cyclicity was introduced recently by Rezaei in Linear Algebra Appl 438(11):4190–4203, (2013). We provide a simple argument, to show that many elements in the commutant of the Volterra operator acting on \(L^p_\mathbb {C}[0,1]\) spaces are not weakly super convex-cyclic.
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Acknowledgments
The authors are indebted to professor Manuel González from University of Cantabria for his interest in our paper and for providing us with a Proof for Lemma 3.
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Communicated by J. Escher.
This paper was partially supported by Junta de Andalucía FQM-257.
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León-Saavedra, F., Piqueras-Lerena, A. Super convex-cyclicity and the Volterra operator. Monatsh Math 177, 301–305 (2015). https://doi.org/10.1007/s00605-015-0750-7
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DOI: https://doi.org/10.1007/s00605-015-0750-7