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THE NORM OF THE PRODUCT OF POLYNOMIALS IN INFINITE DIMENSIONS

Published online by Cambridge University Press:  02 February 2006

C. Boyd  and
R. A. Ryan
C. Boyd
Affiliation:
Department of Mathematics, University College Dublin, Belfield, Dublin 4, Ireland ([email protected])
R. A. Ryan
Affiliation:
Department of Mathematics, National University of Ireland Galway, University Road, Galway, Ireland ([email protected])
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Abstract

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Given a Banach space $E$ and positive integers $k$ and $l$ we investigate the smallest constant $C$ that satisfies $\|P\|\hskip1pt\|Q\|\le C\|PQ\|$ for all $k$-homogeneous polynomials $P$ and $l$-homogeneous polynomials $Q$ on $E$. Our estimates are obtained using multilinear maps, the principle of local reflexivity and ideas from the geometry of Banach spaces (type and uniform convexity). We also examine the analogous problem for general polynomials on Banach spaces.

Keywords

Primary 46B2046G25Secondary 30C1046E50polynomialsgeometry of Banach spacesnorm inequalities
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 49 , Issue 1 , February 2006 , pp. 17 - 28
Copyright
Copyright © Edinburgh Mathematical Society 2006